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of  the 

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This  book  is  DUE  on   the   last  date  stamped  below 


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Form  L-9-2m-7,'22 


«&uprruis'rD  s»tuDv 

EDITED   BY 

ALFRED    L.    HALL-QUEST 


SUPERVISED   STUDY    IN    MATHEMATICS 
AND   SCIENCE 


THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON   •   CHICAGO  •   DALLAS 
ATLANTA  •   SAN   FRANCISCO 

MACMILLAN  &  CO.,  LIMITED 

LONDON  •    BOMBAY  •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


SUPERVISED  STUDY 

IN 

MATHEMATICS  AND  SCIENCE 


BY 

S.   CLAYTON   SUMNER,    M.A. 

SUPERVISING   PRINCIPAL,    PALMYRA,   N.   Y. 
(Formerly  at  Canton,  N.  Y.) 


THE   MACMILLAN   COMPANY 
1922 

All  rights  restrved 


PRINTED  IN  THE  UNITED  STATES  OF  AMERICA 


COPYRIGHT,  1922, 
BY  THE  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.    Published  November,  1922. 


Korfaooti 

J.  S.  Gushing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


So  tbe  flBemors  of 
MY   FATHER 

WHO   BELIEVED   AN   EDUCATION   WAS  THE 

RICHEST  HERITAGE  A  PARENT  COULD 

BEQUEATH    TO    HIS    CHILDREN 


PREFACE 

IN  attempting  a  book  on  supervised  study  which  will  cover 
w*  even  approximately  the  subjects  of  mathematics  and  science, 
,   it  is  impossible  to  do  more  than  give  suggestive  lessons.     This, 
B      therefore,  has  been  my  plan  —  to  give  only  one  or  two  typi- 
cal outlines  of  a  topic  or  subject,  but  to  leave  an  intimation 
of  its  application  whenever  or  wherever  the  teacher  may  elect. 
/J  Thus,  only  one  Red  Letter  Day  lesson  is  presented  in  Algebra, 
*•  but  the  teacher  will  undoubtedly  desire  to  use  many  such 
J  plans  during  the  year.     The  material  in  the  lessons  men- 
tioned may  be  suggestive  for  the  planning  of  others. 

I  have  not  tried  to  add  another  learned  book  in  pedagogy 

?  to  the  many  already  on  the  market.     It  has  rather  been  my 

J  aim  to  write  a  book  that  may  be  of  explicit  and  direct  value 

_      to  the  teacher  or  principal  who  is  daily  striving  to  teach  his 

r*""  children  how  to  study  and  how  to  learn.    I  have  tried  to  write 

it  in  simple  language,  so  that  the  reader  may  get  the  meat, 

if  there  be  any,  without  too  much  stuffing. 

It  is  needless  to  say  that  I  am  a  firm  believer  in  supervised 
study.  It  has  done  much  for  our  children ;  I  am  sure  it  will 
do  more  as  we  progress  in  the  proficiency  of  its  administra- 
tion. It  is  not  a  panacea  for  all  pedagogical  ills,  but  it  is 
valuable  for  what  it  claims  to  be,  and  it  holds  great  promise 
for  the  future. 

I  am  greatly  indebted  to  Professor  Alfred  L.  Hall-Quest 
of  the  University  of  Cincinnati,  who,  as  editor  of  this  series, 
has  not  only  made  it  possible  for  this  volume  to  be,  but  who, 

vii 


viii  Preface 

through  the  reading  and  criticism  of  the  manuscript  and 
through  innumerable  other  suggestions,  has  been  of  ines- 
timable help  to  me. 

Deep  appreciation  is  also  here  expressed  to  Professor 
Charles  M.  Rebert  of  St.  Lawrence  University,  for  valuable 
suggestions  and  advice;  to  Mr.  A.  E.  Breece  of  the  Hughes 
High  School,  Cincinnati,  Ohio,  who  made  a  very  careful  and 
valuable  critical  review  of  the  manuscript  as  relating  to  mathe- 
matics ;  to  my  teachers  at  Canton,  N.  Y.,  who  made  it  possi- 
ble to  actually  try  out  many  of  the  lessons ;  and  to  my  wife, 
for  her  constant  counsel  and  encouragement. 

In  addition,  I  wish  to  acknowledge  my  thanks  for  the  cour- 
tesy of  The  Macmillan  Company,  the  American  Book  Com- 
pany and  the  Charles  E.  Merrill  Company  for  permission 
to  quote  more  or  less  extensively  from  their  publications. 

S.  CLAYTON  SUMNER. 

PALMYRA,  N.  Y. 
January  31, 1923. 


TABLE   OF   CONTENTS 

PAGE 

INTRODUCTION.    Supervised  Study  a  Moral  Imperative   .        .  The  Editor    xiii 

PART  ONE.    MATHEMATICS 

CHAPTER  ONE.    Management  of  the  Supervised  Study  Period  in  Math- 
ematics       3 

FIRST  SECTION.    ALGEBRA  (ELEMENTARY) 

CHAPTER  Two.    Divisions  of  Elementary  Algebra;    Units  of  Instruc- 
tion and  Units  of  Recitation.    A  Time  Table    .        .       20 

ILLUSTRATIVE  LESSONS 

LESSON 

I.    The  Inspirational  Preview 26 

II.     Introduction.     Unit  of  Instruction  I.    A  Lesson  in  Correlation       34 

III.  Introduction  (Continued).     A  How  to  Study  Lesson         .        .      43 

IV.  Introduction  (Continued).    An  Inductive  and  How  to  Study 

Lesson 50 

V.     Addition.     Unit  of  Instruction  IH.     An  Inductive  Lesson: 

Addition  of  Monomials 59 

VI.    Addition   (Continued).    An  Inductive  Lesson:    Addition  of 

Polynomials .      67 

VII.     Simple  Equations.     Unit  of  Instruction  X.    An  Expository 

and  How  to  Study  Lesson :  The  Equation  and  Problems     .       73 

VHI.    Factoring.    Unit  of  Instruction  VII.    A  Socialized  Lesson      .      80 

ix 


Table  of  Contents 


LESSON  PAGE 

IX.     Fractions.    Unit  of  Instruction  IX.     A  Deductive  and  How 

to  Study  Lesson 82 

X.    A  Red  Letter  Day  Lesson 85 

XI.    Radicals.    Unit  of  Instruction  XD7.    A  Socialized  Lesson      .  88 

XII.     Quadratic  Equations.    Unit  of  Instruction  XV.    An  Exposi- 
tory and  How  to  Study  Lesson 92 

XIII.    An  Examination 95 


SECOND  SECTION.    PLANE  GEOMETRY 

CHAPTER  THREE.    Divisions  of  Plane  Geometry;   Units  of  Instruction 

and  Units  of  Recitation 105 

LESSON 

I.    The  Inspirational  Preview 106 

II.    Rectilinear  Figures.    Unit  of  Instruction  II.    A  Deductive 

and  How  to  Study  Lesson :  Vertical  Angles          .        .        .no 

III.  Rectilinear  Figures  (Continued).    A  Deductive  Lesson :   Tri- 

angles   117 

IV.  Rectilinear  Figures  (Continued).    A  How  to  Study  Lesson: 

Originals 123 

V.    Rectilinear  Figures  (Continued).    A  Deductive  Lesson :  Orig- 
inals       129 

VI.    A  Socialized  Review :  Book  I 134 

VII.    An  Exhibition  or  a  Red  Letter  Day  Lesson      .        .        .        .135 


THIRD  SECTION.    ADVANCED  MATHEMATICS 

CHAPTER  FOUR.    Special  Methods  of  Supervised  Study  in  Higher  Math- 
ematics      141 


Table  of  Contents  xi 

PART  TWO.    SCIENCE 

PAGE 

CHAPTER  FIVE.    The  Management  of  the  Supervised  Study  Period  in 

Science 149 

FOURTH  SECTION.    BIOLOGY 

CHAPTER  Six.    Divisions  of  Biology ;  Units  of  Instruction  and  Units  of 

Recitation.    A  Time  Table 155 

A.  —  BOTANY 

LESSON 

I.    The  Inspirational  Preview  156 

II.    Introductory  Topics.     Unit  of  Instruction  I.    A  How  to  Study 

Lesson :  Preliminary  Experiments 161 

III.  Introductory    Topics    (Continued).    An    Inductive    Lesson. 

Problem :  No  Two  Plants  Are  Alike 165 

IV.  Introductory    Topics    (Continued).    An    Inductive    Lesson. 

Problem :  Struggle  for  Existence 169 

V.    Seeds  and  Seedlings.     Unit  of  Instruction  II.    A  How  to  Study 

Lesson:  Seeds  and  Their  Germination         .        .        .        .171 

VI.    Seeds  and  Seedlings  (Continued).    A  Deductive  Lesson :  Lab- 
oratory Experiments 175 

VII.    Seeds  and  Seedlings  (Continued).    A  Socialized  Lesson:    A 

Field  Trip 179 

B.  —  ZOOLOGY 

VIII.    Insects.    Unit  of  Instruction  IX.    A  How  to  Study  Lesson : 

The  Grasshopper 181 

LX.     Insects  (Continued).    A  Laboratory  Lesson:   The  Grasshop- 
per         187 

X.    Insects  (Continued).    A  Correlation  and  Research  Lesson        .     189 

XI.    Insects  (Continued).    A  Socialized  Lesson        ....     192 

XII.    A  Red  Letter  Day  Lesson 195 


xii  Table  of  Contents 

C.  —  PHYSIOLOGY 

LESSON  PAGE 

XIII.     Bones  and  Muscles.     Unit  of  Instruction  XVI.    A  How  to 

Study  Lesson :  Muscles 197 

XTV.     Muscles   (Continued).    A  Laboratory  Lesson  Using  Micro- 
scopic Slides 200 

XV.    Muscles  (Continued).    A  Deductive  Lesson.     Problem:  How 

Muscular  Activity  Is  an  Aid  to  Good  Health        .        .        .     202 

XVI.    Muscles  (Continued).    A  Lesson  in  Correlation        .        .        .     204 
XVII.    An  Examination  Lesson 206 

FIFTH  SECTION.    PHYSICS 
CHAPTER  SEVEN.    Further  Lessons  in  Science 213 

LESSON 

I.     Fluids.     Unit  of  Instruction.    An  Expository  and  How  to 

Study  Lesson 214 

II.     Fluids  (Continued).    A  Laboratory  Lesson       .        .        .        .219 

III.  Fluids  (Continued).    A  How  to  Study  Lesson :  Problems        .     223 

IV.  Red  Letter  Day  Lessons 225 

BIBLIOGRAPHY 229 

INDEX 233 


INTRODUCTION   BY   THE   EDITOR 

SUPERVISED   STUDY  A  MORAL   IMPERATIVE 

MILLIONS  of  words  have  been  written  about  education. 
Theories  have  abounded  and  still  are  fertile.  The  visitor  at 
educational  conventions,  especially  in  the  department  of 
school  superintendents,  is  impressed,  however,  with  the  rapid 
multiplication  of  devices  for  visualizing  educational  prac- 
tice. A  rich  variety  of  moving  picture  machines  and  already 
voluminous  catalogues  of  educational  films  witness  to  the 
dawn  of  a  new  era  in  the  technic  of  teaching.  Later  we  shall 
no  doubt  find  boards  of  censors  passing  upon  these  films  — 
boards  composed  of  theorists,  critic  teachers,  educational 
scientists,  et  al.,  —  but  at  present  the  field  is  open  for  all. 
Doubtless  many  teachers  will  find  in  this  form  of  visual  edu- 
cation an  opportunity  for  enlargement  of  income  as  well  as 
for  the  demonstration  of  teaching  skill. 

Increasing  Emphasis  on  Demonstration.  Demonstration 
and  description  are  rapidly  coming  to  the  front  in  discussions 
of  methods  of  teaching.  One  carefully  prepared  and  suc- 
cessfully performed  demonstration  is  of  more  value  than 
many  verbal  descriptions,  however  clear  these  may  be.  A 
series  of  vivid  verbal  descriptions  makes  definite  and  con- 
crete a  volume  of  abstractions  and  theorizings  on  educational 
practice.  Theory  is  important ;  it  must  not  be  discounted ; 
but  here,  as  elsewhere,  an  illustration  turns  on  light  and  makes 

xiii 


xiv  Introduction  by  the  Editor 

objective  and  easily  understood  the  necessarily  vaguer  dis- 
cussions of  abstract  theory. 

This  series  of  volumes  on  Supervised  Study  attempts  to 
visualize  one  form  of  study  supervision.  Each  book  is  writ- 
ten by  a  teacher  who  has  had  considerable  experience  in  this 
type  of  work.  The  emphasis  in  each  discussion  is  to  make 
concrete  in  as  detailed  description  as  possible,  what  the  author 
has  actually  done  in  his  own  classroom.  Very  briefly  each 
author  states  the  theory  underlying  his  practice,  but  beyond 
this  brief  statement  he  refrains  from  a  discussion  of  principles. 
Teachers  desire  to  see  how  theory  is  applied.  One  cannot  be 
too  clear  and  too  definite  in  describing  the  mode  of  procedure 
in  supervision. 

At  Present  No  Generally  Accepted  Meaning  of  Supervised 
Study.  In  answer  to  those  who  believe  that  Supervised  Study 
as  described  in  this  and  other  volumes  of  the  series  is  different 
from  the  general  understanding  of  the  term,  it  should  be  em- 
phasized that  at  present  there  is  no  generally  accepted  form 
of  Supervised  Study.  It  is  the  conviction  of  the  editor  of 
this  series  that  a  standardized  form  is  undesirable.  The 
main  objective  is  teaching  children  —  all  children  —  how 
to  study  and  guiding  them  while  they  apply  the  principles  of 
correct  studying.  It  is  of  comparatively  little  importance 
how  this  is  done,  providing  it  is  done  effectively.  If  the 
teacher  makes  this  type  of  teaching  superlatively  significant, 
it  follows  that  the  management  of  the  class  and  the  method 
of  presenting  subject  matter  will  change  accordingly.  But 
each  teacher  must  be  the  final  judge  of  how  to  adapt  this  new 
point  of  view  to  local  needs. 

The  Imperative  Need  of  Preventing  Failures  in  School.  It 
should  be  said,  however,  that  any  plan  which  seeks  to  prevent 


Introduction  by  the  Editor  xv 

failures  and  which  aims  to  train  all  pupils  to  study  as  effec- 
tively as  native  ability  permits  is  superior  to  plans  that  simply 
correct  improper  methods  of  work  and  that  are  concerned 
only  with  the  retarded  pupils.  If  school  work  is  limited  to 
the  assigning  and  hearing  of  lessons,  only  a  few  —  the  highly 
endowed  —  will  permanently  profit  by  such  experience. 
There  are  well-meaning  people  who  sincerely  believe  that  the 
school  is  the  place  for  eliminating  society's  mentally  unfit, 
and  that  the  surest  way  of  such  elimination  is  to  assign  les- 
sons, long  and  hard.  Those  who  can  will ;  those  who  cannot 
will  not.  Those  who  will  and  can  are  the  fit ! 

Some  there  are  who  learn  to  swim  by  the  "sink  or  swim" 
method ;  they  are  destined,  forsooth,  to  be  swimmers  if  they 
do  not  sink.  But  how  many  of  you  who  read  these  pages 
learned  to  swim  by  this  fatalistic  method?  There  are  children 
who  early  judge  themselves  incapable  of  school  work.  No- 
body cares !  They  either  can  or  cannot  study.  By  means 
of  the  hard,  soulless  machinery  of  assigning  and  hearing  les- 
sons they  are  cast  out.  We  call  this  a  safe  test  and  out  they 
go  labeled  mentally  weak,  unfit  to  partake  in  a  world  of  thrill- 
ing knowledge,  unfit  to  climb  to  altitudes  of  self-revelation 
and  social  worth.  If,  however,  they  could  have  been  taught 
how  to  use  their  minds,  how  to  partake  in  the  feast  of  knowl- 
edge, who  knows  but  that  many  of  them  would  have  found 
a  new  meaning  of  their  destiny ! 

Supervised  Study  Is  Not  Only  an  Intellectual  Necessity;  It  Is  a 
Moral  Imperative.  As  teachers  it  is  our  plain  duty  to  teach 
children  how  to  study.  The  whole  class  period  must  be  con- 
ducted in  this  spirit.  The  specific  aim  of  every  class  period 
must  be  to  so  direct  the  pupils  that  their  grasp  of  the  new 
work  is  adequate  for  independent  application.  The  teacher 


xvi  Introduction  by  the  Editor 

is  preeminently  a  director  of  study  and  not  primarily  a  dis- 
penser of  subject  matter. 

The  Point  of  View  in  This  Volume.  The  author  of  this 
volume  is  convinced  of  the  effectiveness  of  Supervised  Study. 
He  and  his  teachers  have  tried  it  long  enough  to  know  its 
advantages.  The  subjects  of  mathematics  and  science  are 
especially  favorable  to  this  method.  A  comprehensive  view  of 
the  courses  in  the  high  school  is  given  in  a  series  of  typical 
lessons  describing  in  great  detail  how  children  may  be  directed 
in  beginning,  continuing,  and  reviewing  their  study  of  par- 
ticular units  of  subject  matter.  The  author  is  well  aware  of 
the  movement  for  reorganization  of  courses  especially  in  ninth 
grade  mathematics,  but  inasmuch  as  such  revision  is  not 
likely  to  be  possible  in  all  schools  for  some  tune  to  come  the 
usual  division  of  courses  is  considered  in  this  volume.  It  is 
believed  also  that  even  where  general  mathematics  is  taught, 
not  a  few  pupils  will  elect  additional  special  courses  in  the 
field  of  mathematics.  Inasmuch  as  general  science  is  at 
present  little  more  than  a  combination  of  various  special 
sciences  the  separate  treatment  used  in  this  volume  seems 
preferable.  It  is  hoped  that  general  science  will  evolve  in- 
creasingly along  the  lines  of  natural  correlations  through 
which  the  pupil  will  be  able  to  understand  the  intimate 
relationships  that  exist  among  the  phenomena  of  nature. 


PART   ONE 
MATHEMATICS 


SUPERVISED  STUDY  IN 
MATHEMATICS  AND  SCIENCE 

CHAPTER  ONE 

THE  MANAGEMENT  OF  THE  SUPERVISED  STUDY  PERIOD 
IN  MATHEMATICS 

Causes  of  Failures  in  Mathematics.  —  There  are  a  number 
of  contributory  causes  which,  together  or  separately,  might 
account  for  the  high  mortality  in  mathematics  classes.  That 
it  is  high  is  so  commonly  accepted  among  the  profession,  that 
a  large  percentage  of  failures  has  almost  come  to  be  an  es- 
tablished expectation.  In  eleven  high  schools  near  Chicago, 
the  percentage  of  failures  in  algebra  and  geometry  was  found 
to  be  greater  than  in  any  other  subject.1  In  the  report  of 
the  New  York  State  Education  Department  on  statistics 
for  Regents  Academic  Examinations,  the  failures  in  mathe- 
matics for  the  past  five  years  have  been  between  thirty- 
three  per  cent  and  forty  per  cent.2  The  nearest  competitors 
for  scholastic  dishonors  are  the  commercial  subjects  which 
are  largely  mathematical  in  content. 

The  causes  of  these  failures  are  psychological,  pedagogical, 
and  physical.  Psychologically,  mathematics  has  been  by 

1  School  Review,  June,  1913,  p.  415. 

2  Annual  Report  of  the  State  Department  of  Education  (loth  to  I4th  in- 
clusive), New  York  State. 

3 


4        Supervised  Study  in  Mathematics  and  Science 

almost  common  opinion  accorded  the  position  of  being  the 
hardest  subject  in  the  school  curriculum.  This  estimate  of 
the  subject,  persisted  in  by  pupils,  teachers,  and  the  laity, 
has  inevitably  resulted  in  a  state  of  mind  that  predetermines 
a  large  percentage  of  failures.  Until  we  teachers  succeed  in 
dispelling  this  opinion,  pupils  in  many  instances  will  expect 
to  fail,  and  they  will  fail.  There  is  no  sane  reason  why  mathe- 
matics should  be  so  considered,  and  with  the  new  vision  of 
teaching  the  subject  and  with  the  readjustment  of  the  course 
of  study,  combined  with  its  scientific  treatment  (which  will 
emphasize  the  functional  and  practical  side  instead  of  the 
formal  aspect),  this  view  of  the  severity  of  mathematics 
doubtless  will  gradually  disappear. 

Mathematics  Taught  with  Deliberate  Unattractiveness.  —  It 
is  repeating  a  platitude  to  refer  to  the  fact  that  mathematics 
has  been  very  poorly  taught  in  the  public  school.  There  has 
been  no  serious  lack  of  scholarship  and  of  emphasis  on  the 
acquirement  of  knowledge  of  subject  matter,  but  this  very  em- 
phasis has  tended  toward  the  serious  neglect  of  training  pupils 
to  apply  mathematical  rules  and  formulas  to  practical  reason- 
ing. Too  much  emphasis  has  been  laid  on  the  formal  examina- 
tion, the  "  spectacular  "  effects  according  to  Schultze.1  Too 
much  is  attempted  in  the  time  allotted,  with  insufficient  as- 
similation of  the  matter  studied.  Pupils  are  not  taught 
how  to  study  mathematics.  They  are  only  drilled  on  abstract 
formulas.  The  result  is  an  overdeveloped  memory  and 
undeveloped  powers  of  reasoning. 

Because  of  the  above  noted  unsound  pedagogical  methods, 
with  the  resulting  formal  examinations,  and  because  the 

1  Arthur  Schultze,  "The  Teaching  of  Mathematics  in  Secondary  Schools"; 
The  Macmillan  Company,  1912. 


Supervised  Study  Period  in  Mathematics  5 

pupils  are  graded  chiefly  on  mechanical  ability,  their  prog- 
ress in  mathematics  can  be  determined  to  a  highly  refined 
nicety.  They  have  failed  to  "  do  "  a  certain  number  of  prob- 
lems. Ergo,  they  are  just  that  much  deficient  in  ability  and 
improvement.  There  is  no  leeway  for  difference  of  opinion, 
for  the  exercising  of  the  reasoning  faculty,  for  the  training  of 
individual  characteristics  and  differences.  Being  largely  a 
fact  subject,  as  now  taught,  it  resolves  itself  mainly  into  a 
question  of  "  yes  "  or  "  no,"  and  this  accentuates  the  prob- 
ability of  failure.  Individuals  differ  vastly  in  their  ability  to 
memorize,  and  therefore  the  poor  memorizer  is  placed  at  a 
disadvantage.  The  pupil  who  can  reason  out  a  new  demon- 
stration in  geometry  knows  infinitely  more  geometry  than 
he  who  can  transcribe  on  paper  every  one  of  the  prescribed 
demonstrations  in  a  book  on  this  subject. 

The  Value  and  Place  of  Supervised  Study.  —  This  leads  us 
logically  to  a  discussion  of  the  value  of  supervised  study  in 
mathematics.  Unsupervised  study  is  inefficient  study  be- 
cause much  time  and  energy  are  lost  in  misdirected  effort. 
Pupils  do  not  know  how  to  attack  a  lesson  any  more  than  they 
know  how  to  perform  the  mechanical  processes,  until  they  are 
carefully  taught.  Class  exercises  avail  little  for  the  major- 
ity of  the  pupils  because  no  two  minds  react  in  the  same  way. 
To  clinch  the  class  exercise  individual  guidance  is  required. 
The  unsupervised  recitation  as  a  rule  does  not  provide  for 
this.  Problems  in  algebra  and  originals  in  geometry  are 
entirely  dependent  upon  the  characteristics  of  the  individ- 
ual mind,  which  can  be  developed  and  trained  only  through 
the  individual  himself.  To  quote  from  an  article  by  the 
author,1  "  the  school  must  teach  its  pupils  not  to  be  perfect 

1  Journal  of  the  Neva  York  State  Teachers'  Association,  November,  1918. 


6        Supervised  Study  in  Mathematics  and  Science 

automatons,  responding  with  machine-like  accuracy  to  the 
whim  of  the  examiner,  but  to  become  thinkers,  with  power 
and  knowledge  of  how  to  attack  and  study  out  a  problem, 
how  to  form  personal  opinions,  how  to  get  results,  by  them- 
selves. This,  then,  is  the  function  of  supervised  study:  to 
properly  direct  the  pupil  in  his  work  so  that  he  may  develop 
the  best  methods  of  attacking  problems ;  that  he  may  avoid 
wrong  methods  of  reasoning;  that  he  may  most  efficiently 
employ  his  time ;  and  that  he  may  eventually  acquire  a  power 
of  skill  that  will  classify  him  as  a  finished  thinker,  an  educated 
man." 

Supervised  study  is  only  one  of  the  several  methods  that 
need  to  be  employed  in  bringing  about  a  closer  relationship 
between  teacher  and  pupil  and  in  the  development  of  the 
pupil's  native  endowment  in  the  field  of  mathematics.  Such 
relationship  might  be  illustrated  as  spokes  of  a  wheel.  Just 
as  every  spoke  (Figure  I)  is  necessary  in  the  connection  be- 
tween the  rim  which  represents  the  pupil  and  the  hub  which 
represents  the  teacher,  so  supervised  study  should  be  given 
its  proper  position  in  the  devices  of  the  schoolroom.  The 
other  spokes,  each  with  its  peculiar  evaluation,  might  be  the 
recitation,  the  assignment,  the  equipment,  tests  and  quizzes, 
standard  tests  and  measurements,  inspiration  and  sympathy. 

Division  of  the  Course  into  Units  of  Instruction,  Recitation, 
and  Study.  —  In  our  discussion  of  the  technic  of  the  super- 
vised study  of  mathematics,  let  us  first  agree  on  our  use  of 
terms,  as  formulated  by  Professor  Hall-Quest  in  his  pioneer 
book,  "  Supervised  Study  "  1  and  followed  out  in  the  other  books 
in  this  series.  In  program  of  studies,  let  us  include  all  the 
work  offered  in  a  school ;  by  curriculum,  let  us  understand  a 
1  Hall-Quest,  "Supervised  Study";  The  Macmillan  Company,  1916. 


Supervised  Study  Period  in  Mathematics 


7 


group  of  subjects  leading  to  a  special  end,  as  college  prepara- 
tory curriculum,  domestic  science  curriculum,  etc. ;  and  by 
course,  any  single  subject  as  algebra,  civics,  etc.  Then,  as  a 
means  of  evaluating  the  course  and  giving  it  a  definite  and 


FIGURE  I 

comprehensive  development,  we  shall  separate  the  subject 
matter  into  various  units.  By  units  of  instruction,  we  shall 
mean  the  large  topics  around  which  the  material  revolves. 
In  many  cases  these  are  the  divisions  noted  in  the  table  of 
contents ;  they  are  the  divisions  of  the  subject  into  "  type 


8        Supervised  Study  in  Mathematics  and  Science 

lessons."  l  Such  general  topics  as  percentage,  banking, 
factoring,  graphs,  would  thus  become  units  of  instruction. 
Then  the  subdivisions  of  these  larger  units  into  smaller  ones, 
around  which  one  or  more  recitations  would  revolve,  may  be 
called  units  of  recitation.  These  units  may  again  be  sub- 
divided into  the  work  planned  for  a  single  day,  or  units  of  study. 

Types  of  Recitation.  —  In  addition  to  this  analysis  of  any 
course  of  study  into  its  various  units,  the  careful  teacher  will 
further  decide  on  the  technical  form  of  presentation  of  each 
unit  of  study,  or  the  type  of  lesson.  Following  the  treatment 
of  this  phase  as  detailed  by  Strayer2  and  Earhart,3  we  may 
employ,  as  occasion  prompts,  the  deductive,  inductive,  ha- 
bituation,  expository,  how  to  study,  socialized,  or  review  lessons. 
Since  each  type,  however,  has  its  peculiar  aim  and  technic, 
the  teacher  will  do  well  to  make  a  careful  study  of  them  and 
of  their  application  to  the  subject  in  hand. 

In  general  terms,  the  deductive  lesson  aims  to  draw  forth 
new  conceptions  from  our  present  knowledge.  It  is  based  on 
the  process  by  which  we  think.  The  inductive  lesson,  on  the 
other  hand,  leads  up  to  new  concepts  by  a  series  of  successive 
steps,  each  definite  and  complete  in  itself.  It  is  the  process 
by  which  we  accumulate  knowledge.  In  the  drill  or  habitua- 
tion  lesson,  the  mechanical  side  of  learning  is  stressed.  By 
drill,  needful  automatic  reactions  are  established.  The  ex- 
pository lesson  seeks  to  make  the  new  assignment  as  clear  as 
demonstration  and  analysis  make  it  possible.  It  is  usually 
employed  as  a  connecting  link  between  the  old  and  new  mate- 

1  McMurry,  "How  to  Study";  Houghton  Mifflin  Co.,  1909. 

2  Strayer,  "Brief  Course  in  the  Teaching  Process  ";  The  Macmillan  Company, 
1912. 

3  Earhart,  "Types  of  Teaching";  Houghton  Mifflin  Co.,  1915. 


Supervised  Study  Period  in  Mathematics  9 

rial  of  a  unit  of  study.  The  how-to-study  lesson  is  self-explan- 
atory. Under  review  are  usually  included  written  or  oral 
examinations.  In  a  larger  sense  the  review  should  cover  the 
bringing  together  for  periodic  consideration,  the  clinching  of  a 
unit  of  recitation  or  unit  of  study.  The  socialized  lesson  may 
be  poorly  named  from  a  standpoint  of  nomenclature ;  possibly 
a  better  term  at  the  present  time  would  be  the  democratized 
lesson.  At  any  rate,  it  is  that  type  of  lesson  which  introduces 
the  human  element  into  the  school  work.  By  it  the  work 
outside  and  that  inside  of  the  schoolroom  are  correlated. 
Although  this  type  of  lesson  may  be  used  sparingly,  it  is  none 
the  less  important,  and  the  teacher  should  feel  that  his  greatest 
opportunity  for  reaching  the  consciousness  of  the  child  is 
presented  through  this  form  of  lesson  organization.  Group 
assignments,  dramatic  productions,  class  programs,  dual  proj- 
ects, mathematical  clubs,  and  like  devices  for  impressing  the  in- 
terdependence of  individuals  in  solving  social  or  economic  prob- 
lems, will  tend  to  vitalize  and  democratize  the  subject  matter. 
A  more  elaborate  classification  of  exercise  types  as  applied 
to  the  teaching  of  high  school  mathematics  has  been  evolved 
in  an  illuminating  article  by  Professor  G.  W.  Myers,  of  the 
College  of  Education  of  the  University  of  Chicago,  in  High 
School  Mathematics  and  Science,  June,  1921.  In  this  article, 
the  following  types  of  mathematical  class  exercises  are  dis- 
cussed, and  specifications  or  norms  are  given  for  judging  each : 

I.  The  conceptual  type.  VII.  The  problem  type. 

II.  The  expressional  type.  VIII.  The  topic  type. 

III.  The  associational  type.  IX.  The  applicational  type. 

IV.  The  assimilational  type.  X.  The  test  type. 

V.  The  review  type.  XI.  The  research  type. 

VI.  The  drill  type.  XII.  The  appreciational  type. 


io      Supervised  Study  in  Mathematics  and  Science 

While  space  does  not  permit  a  detailed  review  of  these, 
the  article  in  question  is  commended  to  the  reader  for  care- 
ful study.  To  treat  it  adequately  here  would  be  to  quote  it 
as  a  whole. 

The  Time  Schedule.  —  In  order  that  the  plan  of  supervised 
study  may  be  carried  out  in  its  finest  application,  the  period 
should  be  long  enough  for  the  pupil  to  do  most  if  not  all  of 
his  studying  in  school.  This  would  mean  a  period  of  from 
ninety  minutes  to  two  hours  in  length  and  would  also  involve 
an  extension  of  the  school  day  in  most  cases.  Superintendent 
L.  M.  Allen l  remarks  that  a  shorter  time  than  the  above  is 
"  neither  hay  nor  grass,"  and  that  less  than  forty-five  minutes 
for  the  study  period  itself  will  not  suffice.  The  author  of  this 
book  will  grant  that  the  longer  the  period  the  better  the  results, 
but  from  the  experience  in  his  own  school  for  the  past  five 
years,  he  is  constrained  to  disagree  with  the  above  conclusion 
of  Superintendent  Allen.  In  the  school  at  Canton  the  periods 
are  all  one  hour  in  length,  the  first  thirty-five  minutes  being 
devoted  to  the  recitation  and  the  assignment,  and  the  last 
twenty-five  minutes  being  given  up  to  the  study  of  the  lesson 
for  the  next  day.  Realizing  that  a  longer  study  period  would 
be  very  desirable,  the  author  knows  from  experience  that 
even  this  length  of  time  will  justify  itself  by  increased  and 
better  work,  as  shown  from  the  statistics  as  applied  to  the 
Canton  school. 

It  seems,  therefore,  only  fair  to  conclude  that,  when  it  is 
impossible  to  increase  the  length  of  the  periods  to  the  limits 
suggested  above,  the  installation  of  supervised  study  is  still 
feasible  and  good  results  may  be  secured  from  the  shorter 
period.  In  any  case,  the  period  will  be  divided  into  three 
School  Review,  June,  1917. 


Supervised  Study  Period  in  Mathematics          n 

parts:  the  review,  the  assignment,  and  the  silent  study. 
When  the  periods  are  sixty  minutes  in  length,  the  approximate 
division  of  time  among  these  three  parts  should  be  as  follows : 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Silent  Study 25  minutes 

With  a  longer  period,  the  study  section  will  be  increased 
more  than  the  others.  A  sixty  minute  period  means,  of  course, 
that  only  part  of  the  study  will  be  done  in  class;  a  ninety 
or  one  hundred  twenty  minute  period  should  make  home 
work  unnecessary. 

The  Review.  —  The  review  will  take  the  place  of  the  old 
recitation  and,  while  its  length  has  been  decreased,  by  inten- 
sive and  well-applied  questions  the  work  ought  to  be  thor- 
oughly covered  in  this  time.  The  class  review  should  es- 
sentially be  a  re-view  of  the  difficult  parts  of  the  day's  lesson ; 
and,  while  the  weaker  pupils  will  get  the  most  attention,  it 
will  be  a  sort  of  summing  up  for  all.  It  is  unnecessary  to  re- 
view every  minute  step  ;  half  of  the  usual  recitation  is  spent  in 
reciting  on  perfectly  well-known  and  understood  things.  The 
review  is  not  the  time  to  show  of  what  we  know  but  to  clear  up  the 
things  we  do  not  know  or  know  only  indistinctly.  It  should 
always  be  a  real  step  forward,  a  sort  of  clearing  house  for  the 
previous  day's  assignments.  Again,  the  usual  type  of  recitation 
is  apt  to  be  a  kind  of  monologue  with  the  teacher  taking  the 
leading  part.  As  a  matter  of  fact,  the  teacher  should  remain 
in  the  background.  In  war,  the  generals  give  orders  but  the 
rank  and  file  does  the  fighting.  The  review,  then,  should  be 
incisive,  intensive,  and  conclusive.  The  entire  class  should 
not  be  held  back  by  a  few  backward  pupils.  On  the  other 


12      Supervised  Study  in  Mathematics  and  Science 

hand,  the  bright  pupils  should  not  be  conciliated  by  insipid 
questions.  The  teacher  should  address  the  review  to  the 
weaker  ones  but  by  methods  that  will  appeal  to  all. 

The  Assignment.  —  The  assignment  is  always  the  most 
important  part  of  the  recitation  period.  It  should  include, 
in  addition  to  a  definite  allotment  of  new  work,  very  clear 
explanations.  The  advance  lesson  should  be  carefully  planned 
beforehand,  so  that  there  will  be  a  definite  amount  of  ground 
to  be  covered,  a  definite  objective  gained,  and  a  definite  advance 
made.  If  this  is  slurred  over,  the  pupil  will  have  no  clear 
idea  of  what  the  lesson  is  about  or  of  what  he  is  to  do.  The 
assignment  to  be  prepared  should  also  be  made  with  due 
consideration  of  its  difficulty,  the  varying  abilities  of  the 
members  of  the  class,  and  the  often  overlooked  fact  that  the 
pupils  have  lessons  also  in  other  departments.  If  all  these 
elements  have  been  taken  into  consideration  and  given  thought- 
ful planning,  the  time  allotted  to  this  particular  section  of  the 
period  ought  to  be  sufficient.  But  there  must  be  plenty  of 
time  to  cover  the  assignment  fully  and  thoroughly;  there- 
fore, it  is  better  to  assign  too  little  than  too  much.  In  any 
quota  of  problems  or  examples,  special  difficulties  likely  to  be 
encountered  should  be  pointed  out  and  possible  methods  of 
attack  suggested.  The  assignment  is  perfect  only  when  every 
pupil  knows  exactly  what  is  the  aim  of  the  new  work,  what 
is  the  best  method  for  its  solution,  and  just  what  ground  he 
is  expected  to  cover  before  the  next  recitation. 

The  Study  Period.  —  The  study  section  is  the  part  devoted 
by  the  pupil  to  the  study  of  the  advance  lesson,  under  the 
direct  and  sympathetic  supervision  of  the  teacher.  In  the 
sixty  minute  period  used  in  the  author's  school,  the  pupils 
are  not  all  expected  to  complete  the  assignment  during  the 


Supervised  Stttdy  Period  in  Mathematics          13 

period.  Some  will,  however,  and  it  will  be  an  incentive  to 
all  to  strive  to  complete  the  work  during  the  time  allotted  to 
study.  At  any  rate,  all  will  have  been  able  to  get  a  start  and 
a  start  in  the  right  direction. 

The  teacher  will  have  two  classes  of  pupils  to  look  after 
during  the  silent  study  period.  One  will  be  those  who  have 
some  little  technical  difficulty  with  the  new  lesson.  A  well- 
directed  question  will  usually  set  them  on  the  right  path. 
The  other  division  will  be  those  who  are  commonly  considered 
failures,  but  who  in  many  instances  are  simply  pupils  with 
some  individual  characteristics  which  react  unfavorably  for 
maximum  efficiency.  These  pupils  should  be  carefully  studied 
by  the  teacher  and  their  cases  diagnosed.  The  problems 
thus  presented  to  the  teacher  should  awaken  all  of  his  deter- 
mination to  solve  them.  The  silent  study  period  thus  gives 
the  teacher  an  opportunity  for  a  study  of  the  individual 
personnel  of  his  class.  The  elimination  of  pupils  from  the 
list  of  failures  should  become  the  predominant  effort  rather  than 
the  elimination  of  pupils  from  class  and  school.  Often  a  pupil, 
who  would  ordinarily  have  failed,  has  found  himself  through 
a  little  attention  and  study  on  the  teacher's  part  during  the 
study  period.  When  the  teacher  can  sit  down  with  him,  note 
his  manner  of  work,  detect  his  deficiences  and  weaknesses, 
and  by  a  little  tactful  and  sympathetic  guidance,  lead  him 
into  the  paths  of  success,  such  a  pupil  will  gain  confidence  and 
later  economical  independence.  He  must  be  taught  how  to 
walk  —  how  to  study. 

The  Assignment  Sheet.  —  The  assignment  sheet  used  in 
Canton,  which  is  similar  to  the  one  described  by  Miss  Simpson 
in  her  companion  book  in  this  series,1  is  reproduced  at  the  end 
1  "  Supervised  Study  in  History" ;  The  Macmillan  Company,  1918. 


14     Supervised  Study  in  Mathematics  and  Science 

of  this  chapter.  The  object  of  the  sheet  is  to  induce  the 
teacher  to  have  a  definite  plan  for  each  day's  work.  Work  not 
carefully  planned  is  apt  to  be  poorly  done.  Nothing  begets 
carelessness  and  indifference  on  the  part  of  the  class  so  much 
as  a  lack  of  purpose  and  plan  on  the  part  of  the  teacher. 
These  sheets  need  take  scarcely  any  time ;  in  fact,  they  will 
save  time,  because  the  teacher  will  know  exactly  what  he  is 
going  to  do,  what  material  he  is  going  to  use,  and  where  it  is 
to  be  found.  In  addition  thereto,  he  will  have  at  the  close 
of  the  term  a  complete  record  of  work  accomplished. 

How  to  Make  a  Lesson  Plan  Sheet.  —  Under  Review  note 
exactly  the  things  that  need  to  be  reemphasized.  The  ref- 
erences to  other  books  for  supplementary  material  may  be 
noted  under  Memoranda.  A  good  scheme  is  to  write  on  the 
back  of  the  assignment  sheet  the  names  of  the  pupils  who 
should  receive  especial  attention  during  this  review. 

The  Threefold  Assignment.  —  As  Professor  Hall-Quest 
explains  in  his  book,  the  assignment  should  be  in  three  parts : 
one  to  take  care  of  the  inferior  pupil,  one  to  take  care  of  the 
average  pupil,  and  one  to  take  care  of  the  superior  pupil. 
Hence  we  have  the  three  part  assignments,  or  the  minimum- 
.  average-maximum  plan. 

The  minimum  assignment  should  cover  the  minimum  es- 
sentials, i.e.  the  work  that  all  must  do  at  the  very  least,  and 
that  the  majority  can  easily  do  within  the  twenty-five  minutes 
allotted  to  study.  It  should  be  so  planned  that  pupils  who 
never  do  any  more  than  this  amount  will  be  able  to  pass  the 
final  examination,  which  is  or  should  be  the  minimum  require- 
ment, but  too  often  becomes  the  only  aim  of  the  teacher. 
These  pupils  will  not  obtain  a  high  mark,  but  they  will  have 
mastered  enough  of  the  subject  to  get  a  passing  grade.  The 


Supervised  Study  Period  in  Mathematics          15 

object  of  the  average  and  maximum  assignments  is  to  produce 
pupils  who  will  not  only  pass  but  pass  high. 

The  average  assignment  should  include  more  examples  and 
different  kinds  of  examples  but  not  necessarily  of  a  much 
harder  nature.  However,  the  more  problems  a  pupil  can 
solve  correctly  in  a  given  time,  the  more  skilled  he  will  be- 
come, up  to  a  certain  limit. 

The  maximum  assignment  is  to  take  care  of  the  brighter 
pupils,  —  those  who  are  able  to  do  more  than  the  average 
pupil  and  who  should  have  some  incentive  to  do  advanced 
work.  Usually  this  assignment  will  be  given  from  other  texts 
and  will  consist  of  more  difficult  material.  This  section 
should  be  so  limited  in  amount  as  not  to  discourage  but  to 
incite  to  a  desire  to  cover  it.  It  will  incidentally  keep  a 
disturbing  pupil  out  of  mischief.  But  the  tasks  should  be 
constructive  and  not  simply  the  old  fashioned  "  busy  "  work 
which  employs  but  does  not  develop. 

Under  Study  note  any  points  to  be  kept  in  mind  while  the 
pupils  are  working,  especially  in  regard  to  mistakes  they  are 
likely  to  make.  This  section  of  the  sheet  may  be  made  very 
effective,  if  the  teacher  is  fully  alive  to  the  function  of  the 
supervised  study  idea.  In  the  course  of  a  year  this  plan  will 
have  become  a  real  series  of  methods. 

The  loose-leaf  notebook  makes  an  excellent  method  of 
preserving  the  sheets.  They  will  be  found  of  inestimable 
value  another  semester. 

How  to  Use  the  Assignment  and  Study  Sheet.  —  Do  not 
become  a  slave  to  the  sheet;  make  it  your  servant.  If  it 
means  omitting  your  daily  recreation,  eliminate  it,  not  your 
recreation.  It  ought  not  to  take  much  time  and,  as  has  been 
suggested,  it  will  eventually  be  found  to  save  time. 


1 6      Supervised  Study  in  Mathematics  and  Science 

Strive  always  to  inspire  the  class  to  reach  the  maximum 
assignment,  to  raise  the  maximum  and  lower  the  minimum  end. 

Determine  the  proper  proportions  of  pupils  who  should  get 
the  various  sections.  The  following  is  a  normal  distribution : 
80  per  cent  should  complete  the  average  assignment,  10  per 
cent  the  minimum  only,  and  10  per  cent  the  maximum. 
When  these  percentages  vary  greatly  from  the  established 
norm,  the  nature  and  length  of  the  assignments  should  be 
modified  proportionately.  That  is,  in  a  class  of  30  pupils,  24 
should  do  all  the  average  assignment  before  the  end  of  the 
period.  Three  will  be  behind  and  will  need  special  attention 
next  day ;  three  will  be  working  some  of  the  examples  in  the 
maximum  assignment.  This  of  course  is  subject  to  varia- 
tion from  day  to  day,  but  may  serve  as  a  guide. 

At  the  close  of  the  period,  the  teacher  should  ascertain  how 
many  of  the  class  have  completed  the  minimum,  the  average, 
and  the  maximum  assignments.  This  may  be  done  by  having 
all  hand  in  their  papers  a  few  seconds  before  the  period 
ends.  The  pupil  may  note  on  his  paper  which  assignments 
he  has  completed.  What  remains  of  the  assignment  may  be 
required  of  the  pupils  the  next  day.  Various  schemes  may 
be  evolved ;  some  will  be  touched  upon  in  connection  with  the 
illustrative  lessons  in  Part  One. 

The  assignment  should  be  written  each  day  upon  the  board 
and  numbered  I,  II,  III  to  correspond  with  the  different 
assignment  groups.  The  pupils  need  not  be  told  the  reason 
for  this  as  it  will  be  better  if  they  do  not  understand  the 
reason  for  the  differentiation.  The  pupils  should  form  the 
habit  of  noting  down  the  assignment,  thus  getting  practice 
in  keeping  a  written  record  of  important  facts  and  engage- 
ments. 


Supervised  Study  Period  in  Mathematics          17 

The  Management  of  the  Supervised  Study  Period.  —  As 
soon  as  the  class  meets,  take  the  roll  call  and  at  once  start 
the  review.  The  time  spent  in  taking  the  roll  should  be 
reduced  to  a  minimum ;  certainly  not  over  two  minutes  at  the 
most.  An  excellent  method  is  to  have  on  a  sheet  of  paper  a 
diagram  representing  the  seats  and  in  each  space,  which 
represents  a  seat,  put  the  name  of  the  pupil  who  occupies  it 
each  day.  Then  those  absent  can  be  quickly  noted  through 
the  vacant  spaces,  and  their  names  checked. 

The  time  allotted  to  the  review  and  the  assignment  during 
the  first  thirty-five  minutes  should  be  held  to  as  closely  as 
possible ;  a  schedule  or  time  table  is  absurd  unless  adhered  to 
conscientiously.  At  the  end  of  the  thirty-five  minutes,  a 
bell  may  be  rung  simultaneously  in  all  rooms  from  the  study 
hall  or  some  other  central  place,  and  the  next  two  minutes 
devoted  to  physical  drills,  setting-up  exercises,  etc. 

Then,  for  the  next  twenty-five  minutes  or  fraction  thereof, 
the  pupils  should  be  required  to  work  on  their  new  lesson. 
The  teacher  should  insist  that  no  other  work  be  done  during 
the  study  period,  until  the  lesson  in  hand  is  finished.  At 
first,  some  may  try  to  study  the  succeeding  lesson  but  a  little 
firmness  will  soon  bring  desired  results. 

As  soon  as  possible  after  the  class  has  been  organized,  it 
may  be  expedient  to  reseat  the  pupils  according  to  their  abili- 
ties ;  those  of  minimum  ability  on  one  side  of  the  room,  those 
of  maximum  ability  on  the  opposite  side,  and  the  average 
pupils  between.  This  allows  the  teacher  to  come  into  closer 
contact  with  the  weaker  pupils,  and  he  can  give  them  addi- 
tional attention.  This  classification  must  be  done  with  tact 
so  as  not  to  hurt  any  child's  feelings ;  therefore,  the  earlier  in 
the  term  it  is  done  the  better.  It  is  not  necessary  to  name 


1 8      Supervised  Study  in  Mathematics  and  Science 

the  groups.     Calling  them  A,  B,  C  without  any  further  char- 
acterization will  suffice. 

Some  method  of  checking  the  results  of  the  work  accom- 
plished during  the  period  should  be  worked  out.  Various 
schemes  will  be  mentioned  in  connection  with  the  illustrative 
lessons ;  but  a  few  general  remarks  may  be  made  here.  Any 
plan  which  would  depend  entirely  on  the  amount  of  work  done 
in  class  must  take  into  consideration  the  fact  that  some 
children  are  accurate  but  slow ;  these  must  not  be  discouraged 
by  low  marks.  While  rapidity  is  desired,  accuracy  is  more 
important,  and  these  pupils  should  be  encouraged  to  make  an 
effort  to  maintain  accuracy  and  secure  greater  rapidity.  On 
the  other  hand,  it  is  very  important  that  as  much  of  the  work 
be  done  in  class  as  practicable  so  that  the  teacher  may  know 
that  the  work  is  the  pupil's  own.  Thus  there  is  presented  to 
the  teacher  a  fine  question  for  analysis :  to  discover  why  a 
pupil  does  not  accomplish  so  much  as  is  expected.  The  teacher 
should  be  untiring  in  his  effort  to  solve  such  a  problem. 
Theoretically  the  pupils  who  habitually  solve  only  the  first 
or  minimum  assignment  should  receive  a  mere  passing  grade, 
those  doing  the  average  assignment  should  receive  marks 
between  80  and  90,  and  those  doing  the  maximum  amount 
should  receive  honors.  But  this  must  eventually  be  deter- 
mined by  the  ability  of  the  pupil  to  work  similar  exercises  in 
the  review,  by  his  lack  of  dependence  upon  the  teacher  during 
the  study  period,  and  to  a  lesser  extent  by  the  periodic  test. 


Supervised  Study  Period  in  Mathematics          19 

ASSIGNMENT  AND   STUDY  SHEET 

SUBJECT PERIOD .DATE 

UNIT  or  INSTRUCTION 

UNIT  OF  RECITATION 

UNIT  OF  STUDY 

LESSON  TYPE.... 


REVIEW  : 

MEMORANDA 

ASSIGNMENT  : 

i.   MINIMUM 

2.  AVERAGE 

3.  MAXIMUM 

STUDY: 

Number  of  pupils  solving  minimum  assignment- 
Number  of  pupils  solving  average  assignment.... 
Number  of  pupils  solving  maximum  assignment. 


Total. 
FIGURE  II 


FIRST  SECTION.    ALGEBRA 
CHAPTER  TWO 

DIVISIONS   OF  ELEMENTARY  ALGEBRA 

It  is  advisable  in  any  course  of  study  that  the  teacher  have 
a  definite  outline  of  the  successive  stages  in  the  development 
of  the  subject,  and  that  these  be  further  subdivided  into  their 
smaller  units.  These  general  topics  may  be  called  Units  of 
Instruction  and  the  subtopics,  Units  of  Recitation.  The 
course  of  study  here  outlined  is  that  suggested  by  the  New 
York  State  Education  Department  in  its  1910  syllabus. 

A.  UNITS  OF  INSTRUCTION. 

I.  Introduction. 

II.  Positive  and  negative  numbers. 

III.  Addition. 

IV.  Subtraction. 
V.  Multiplication. 

VI.  Division. 

VII.  Factoring. 

VIII.  Common  factors  and  multiples. 

IX.  Fractions. 

X.  Simple  equations. 

XI.  Graphic  representation. 

XII.  Involution. 

XIII.  Evolution. 

XIV.  Radicals. 

XV.  Quadratic  equations. 

B.  UNITS  OF  RECITATION. 

20 


Divisions  of  Elementary  Algebra  21 

The  subdivisions  do  not  imply  that  only  one  recitation  is  to  be 
given  to  each  topic,  but  rather  that  all  the  recitations  for  the  length 
of  time  needed  will  center  around  this  special  topic. 

I.  INTRODUCTION.    The  material  given  under  this   head 
varies  in  different  texts,  but  it  will  at  least  contain : 

Units  of  Recitation: 

1.  Symbols  of  algebra. 

2.  Literal  numbers. 

3.  Historical  notes. 

4.  Definitions  and  notation. 

II.  POSITIVE  AND  NEGATIVE  NUMBERS. 

Units  of  Recitation : 

1.  Explanation  and  illustration  of  signed  numbers. 

2.  The    addition,    subtraction,    multiplication,   and 

division  of  signed  numbers. 

HI.  ADDITION. 

Units  of  Recitation: 

1.  Addition  of  monomials. 

2.  Addition  of  polynomials. 

IV.  SUBTRACTION. 

Units  of  Recitation : 

1.  Subtraction  of  monomials. 

2.  Subtraction  of  polynomials. 

3.  The  parenthesis. 

V.  MULTIPLICATION. 
Units  of  Recitation: 

1.  Multiplication  of  monomials  by  monomials. 

2.  Multiplication  of  polynomials  by  monomials. 

3.  Multiplication  of  polynomials  by  polynomials. 

4.  Special  cases. 


22     Supervised  Study  in  Mathematics  and  Science 

VI.  DIVISION. 

Units  of  Recitation : 

1.  Division  of  monomials  by  monomials. 

2.  Division  of  polynomials  by  monomials. 

3.  Division  of  polynomials  by  polynomials. 

VII.  FACTORING. 

Units  of  Recitation : 

1.  To  factor  a  monomial. 

2.  To  factor  a  polynomial. 

3.  To   factor   a    polynomial   whose   terms   may   be 

grouped  to  show  a  common  polynomial  factor. 

4.  To  factor  a  trinomial  which  is  a  perfect  square. 

5.  To  factor  the  difference  of  two  squares. 

6.  To  factor  a  trinomial  in  the  form  of  x2  +px+q. 

7.  To  factor  a  trinomial  in  the  form  of  axz+bx+c. 

8.  To  factor  the  sum  or  difference  of  cubes. 

9.  To  factor  the  sum  or  difference  of  the  same  odd 

powers  of  two  numbers. 

10.  To  factor  the  difference  of  the  same  even  powers 

of  two  numbers. 

11.  To  factor  by  the  factor  theorem. 

12.  To  factor  by  special  devices. 

VIII.  COMMON  FACTORS  AND  MULTIPLES. 

Units  of  Recitation : 

1.  Highest  common  factor. 

2.  Least  common  multiple. 

IX.  FRACTIONS. 

Units  of  Recitation : 

1.  Reduction  to  higher  or  lower  terms. 

2.  Reduction  to  an  integral  or  mixed  expression. 

3.  Reduction  to  similar  fractions. 


Divisions  of  Elementary  Algebra  23 

4.  Addition  of  fractions. 

5.  Subtraction  of  fractions. 

6.  Multiplication  of  fractions. 

7.  Division  of  fractions. 

8.  Complex  fractions. 

X.  SIMPLE  EQUATIONS. 
Units  of  Recitation : 

1.  Equations  with  one  unknown. 

2.  Equations  involving  fractions. 

3.  Literal  equations. 

4.  Problems. 

5.  Simultaneous  equations. 

a.  Elimination  by  addition  or  subtraction. 

b.  Elimination  by  comparison. 

c.  Elimination  by  substitution. 

6.  Literal  simultaneous  equations. 

7.  Problems. 

8.  Equations  with  three  or  more  unknowns. 

9.  Problems. 

XI.  GRAPHIC  REPRESENTATION. 
Units  of  Recitation : 

1.  Graphs  of  statistics. 

2.  Graphic  representation  of  linear  equations. 

XII.  INVOLUTION. 

Units  of  Recitation: 

1.  Involution  of  monomials. 

2.  Involution  of  polynomials. 

3.  Involution  by  the  binomial  theorem. 

XIII.  EVOLUTION. 

Units  of  Recitation : 

1.  Evolution  of  monomials. 

2.  To  extract  the  square  root  of  polynomials. 


24     Supervised  Study  in  Mathematics  and  Science 

XIV.  RADICALS. 

Units  of  Recitation: 

1.  To  reduce  radicals  to  their  simplest  form. 

2.  To  reduce  a  mixed  surd  to  an  entire  surd. 

3.  To  reduce  radicals  to  the  same  order. 

4.  Addition  and  subtraction  of  radicals. 

5.  Multiplication  of  radicals. 

6.  Division  of  radicals. 

7.  Involution  and  evolution  of  radicals. 

8.  Rationalization. 

9.  Square  root  of  a  binomial  quadratic  surd. 
10.  Radical  equations. 

XV.  QUADRATICS. 

Units  of  Recitation: 

1.  Pure  quadratic  equations. 

2.  Affected  quadratic  equations. 

3.  Solution  by 

a.  Factoring. 

b.  Completing  the  square. 

c.  The  formula. 

4.  Literal  quadratic  equations. 

5.  Radical  equations  in  quadratics. 

6.  Problems. 

7.  Simultaneous  equations  involving  quadratics. 

a.  One  simple  equation,  one  involving  the  second 

degree. 

b.  Two    homogeneous    equations    of    the    second 

degree. 

c.  Symmetric  equations  of  third  or  fourth  degree, 

readily  solvable  by  dividing  the  variable 
member  of  one  by  the  variable  member  of 
the  other. 

8.  Problems. 


Divisions  of  Elementary  Algebra  25 

Time  Table  for  the  Term.  —  Below  is  a  suggested  time 
table  for  the  year's  work : 

ALGEBRA :    40  weeks 
FIRST   TERM 

ist  week Introduction 

2d  week Addition 

3d,  4th,  5th  weeks Subtraction  and  parenthesis 

6th  and  ?th  weeks Multiplication 

8th  and  gth  weeks Division  and  review 

loth  to  isth  week Factoring 

1 6th  week Common  factors  and  common  multiples 

iyth  to  2oth  week Fractions 

SECOND    TERM 

aist  to  24th  week Simple  equations 

25th  week Graphs 

a6th  and  27th  weeks Involution 

28th  and  2gth  weeks Evolution 

3oth  to  34th  week Radicals 

35th  and  36th  weeks Quadratics 

37th  to  4<Dth  week Review 

Factors  Modifying  the  Foregoing  Arrangements.  —  Local 
conditions  will  necessarily  determine  the  emphasis  to  be  placed 
on  the  respective  units  of  subject  matter.  If  a  course  in 
general  mathematics  is  followed,  it  is  obvious  that  considerable 
modifications  will  be  required.  Some  classes  are  more  mathe- 
matically-minded than  others  and  this  fact  should  affect 
emphasis.  Whatever  units  are  found  to  be  rarely  of  value 
even  in  advanced  studies  of  mathematics  should  be  practically 
ignored.  There  is  not  time  in  public  school  work  for  the 
elaboration  of  the  useless.  Many  textbooks  in  mathematics 
are  now  so  arranged  that  certain  designated  sections  may  be 
omitted  without  affecting  the  continuity. 


26     Supervised  Stiidy  in  Mathematics  and  Science 

LESSON  I 

THE  INSPIRATIONAL  PREVIEW 

Purpose.  —  The  purpose  of  such  a  lesson  is  to  arouse  in  the 
child  the  will  to  learn,  to  awaken  interest  in  the  study  of 
algebra,  and  to  outline  in  a  general  way  some  of  the  things  of 
interest  which  will  be  studied  during  the  course. 

Need.  —  Coming  from  the  grades  with  no  very  clear  idea  of 
what  high  school  means  and  assigned  to  new  subjects,  the 
very  names  of  which  are  often  strange,  it  is  no  wonder  that  the 
young  pupil  is  not  only  lacking  in  any  particular  interest  in 
his  new  work,  but  may  even  have  a  natural  antipathy  toward 
it  from  the  start,  unless  interest  be  aroused  through  this 
inspirational  preview. 

It  is  important  in  meeting  any  class  for  the  first  time  that 
the  teacher  get  en  rapport  with  the  pupil  as  quickly  as  possible. 
It  is  well  in  place  to  give  a  simple  talk  on  the  history,  practical 
value,  and  general  content  of  the  subject,  —  a  sort  of  advertis- 
ing or  "  selling"  talk.  The  careful  traveler  will  always  plan 
his  trip  ahead  in  order  that  he  may  be  prepared  to  note  all  the 
important  and  interesting  things  that  may  lie  in  store  for  him ; 
otherwise,  many  things  would  escape  his  attention.  So  the 
preview  is  a  sort  of  bird's-eye  view  of  the  course,  a  cranking 
up  of  the  pupil's  interest,  preparatory  to  a  good  start  and  a  run. 

Method.  —  Simple  language  should  be  used.  The  class  is 
composed  of  immature  boys  and  girls  to  whom  big  words  and 
phrases  mean  little ;  the  talk  should  be  more  to  the  child 
than  about  the  subject.  The  essential  thing  is  to  make  the 
children  feel  at  home,  to  arouse  enthusiasm  for  the  subject, 
and  to  make  them  look  forward  to  their  work  in  algebra  with 
pleasure. 


Divisions  of  Elementary  Algebra  27 

When  a  strange  word  or  important  date  is  given,  it  will  have 
more  effect  upon  the  class  if  it  is  written  upon  the  blackboard 
at  the  time  that  it  is  mentioned.  Prearranged  work  upon 
the  board,  simply  referred  to  in  passing,  does  not  rivet  their 
attention  so  well. 

When  the  preview  is  completed,  it  might  be  well  to  ask  a  few 
questions  concerning  what  has  been  said  and  give,  if  neces- 
sary, any  further  details  needed  to  make  the  subject  clear. 
Ask  the  pupils  to  jot  down  the  data  you  have  placed  upon 
the  board  and  tell  them  to  hand  in  the  next  day  a  simple  state- 
ment of  what  has  been  said.  This  is  not  so  important  from 
the  standpoint  of  the  material  as  it  is  from  the  standpoint  of 
inculcating  at  once  the  necessity  of  paying  attention,  of  being 
specific  regarding  facts ;  and  of  the  implication  that  mathe- 
matics is  closely  correlated  with  English  in  the  clearness  of 
its  exposition. 

Historical.  —  A  little  of  the  history  of  the  subject  will 
arouse  the  pupils'  interest  in  the  age  and  romance  of  algebra. 
It  is  best  not  to  go  much  into  detail  because  of  the  complexity 
of  its  historical  development.  It  will  be  better  also  to  intro- 
duce each  new  topic,  when  studied,  by  its  individual  history. 
Many  of  the  recent  texts  in  algebra  have  historical  notes 
scattered  through  the  book  as  a  help  to  humanizing  the  sub- 
ject. Pictures  of  famous  mathematicians  add  to  the  interest 
of  these  historical  references.  A  framed  portrait  of  one  or 
two  mathematicians  of  note,  or  a  statuette,  placed  in  the  room, 
will  give  an  added  atmosphere  to  the  study  of  the  subject. 

Origin  of  the  Word  "Algebra."  Its  name  is  derived  from  the 
title  of  a  book  which  the  Arabs  introduced  into  Europe  in  the 
ninth  century.  The  full  title  of  the  book  was  "  Al-jebr  w'  al 
muqabalah,"  of  which  the  first  two  syllables  have  been  cor- 


28     Supervised  Study  in  Mathematics  and  Science 

rupted  into  the  present  term,  algebra.  In  the  original  tongue 
it  referred  to  the  process  of  transposing  terms  in  the  manipu- 
lation. Hence  the  solving  of  problems  was  early  considered 
the  main  business  of  this  subject. 

Contributors  to  the  Subject.  Modern  algebra  has  not  come 
down  to  us  in  its  present  composition,  but  like  the  automobile 
and  every  other  invention,  it  is  the  result  of  years  of  growth 
and  of  contributions  of  many  minds.  The  early  Egyptians 
and  the  ancient  Greeks  of  the  "golden  age"  had  some  con- 
ception of  the  equation  and  have  left  their  imprint  upon  its 
development.  Heron  of  Alexandria  about  100  B.C.  made  the 
greatest  advance  in  its  development  up  to  that  time,  but 
Diophantus,  a  fourth-century  Greek,  was  the  first  to  write  an 
entire  book  upon  the  subject.  He  emphasized  indeterminates, 
which  are  even  now  called  Diophantine  after  him.  But  as 
stated  above,  the  book  whose  title  has  given  the  subject  its 
modern  name  was  the  first  general  treatise  of  importance. 
The  modern  founder  of  algebra  was  a  Frenchman,  named 
Vieta,  who,  in  1591,  gave  the  science  the  technical  symbolism 
which  is  in  use  to-day.  Among  other  modern  contributors 
were  Wessel  of  Norway,  Gauss  of  Denmark,  and  Sir  Isaac 
Newton  of  England. 

Interesting  Incidents.  The  history  of  the  subject  abounds 
in  many  interesting  episodes.  It  is  said1  that  Sir  William 
Hamilton,  an  Englishman,  who  had  been  working  for  years  on 
a  certain  problem,  was  one  day  taking  a  stroll  with  his  wife, 
when  the  solution  suddenly  flashed  into  his  mind,  and  he  at 
once  engraved  upon  a  stone  in  Brougham  Bridge,  which  he 
was  crossing  at  the  time,  one  of  the  fundamental  formulas  of 

1 " Science-History  of  the  Universe,"  VIII;  Mathematics,  Current  Litera- 
ture Publishing  Company,  1912. 


Divisions  of  Elementary  Algebra  29 

modern  algebra,  called  quaternions.  This  bridge  has  ever 
since  been  called  Quaternion  Bridge. 

An  Italian  named  Tartaglia,  who  claimed  a  certain  alge- 
braic discovery,  was  challenged  by  another  famous  mathe- 
matician by  the  name  of  Fiori.  The  contest  was  to  see  which 
one  could  solve  the  greatest  number  of  a  collection  of  thirty 
problems  within  thirty  days.  Tartaglia,  by  using  his  new 
discovery,  which  by  the  way  is  now  a  matter  of  common 
knowledge,  i.e.  the  cubic  equation,  solved  the  entire  thirty 
within  two  hours'  time.  He  celebrated  his  triumph  by  com- 
posing some  verses,  but,  according  to  the  custom  of  the  times, 
he  kept  his  discovery  a  secret  for  many  years. 

Practical  Value  of  Mathematics  in  General  and  Algebra  in 
Particular.  —  From  the  very  earliest  times,  the  study  of 
mathematics  has  been  considered  primarily  practical.  Mathe- 
matics has  been,  in  some  form  or  other,  the  bulwark  of  the 
education  of  the  Chinese,  the  Arabian,  the  Assyrian,  the  Jew, 
the  Greek,  the  Roman,  and  every  modern  race.  One  might 
as  well  try  to  conceive  of  life  without  atmosphere  as  to  try  to 
separate  the  influence  of  mathematics  from  human  life  and 
endeavor. 

Dr.  Eugene  Smith  of  Columbia  University  says  that  "  if 
all  mathematical  knowledge  were  eliminated,  civilization 
would  be  demoralized,  factories  would  stop  for  want  of  ma- 
chinery, and  life  would  revolt  to  chaos." * 

This  is  the  era  of  machinery.  A  machine,  mathematically 
wrong,  is  a  failure.  Long  before  the  machine  is  put  on  the 
market,  however,  it  has  been  the  subject  of  painstaking  effort 
on  the  part  of  the  inventor,  the  draftsman,  the  pattern- 

1  "Mathematics  in  Training  for  Citizenship";  Teachers  College  Record, 
May,  1917. 


3o      Supervised  Study  In  Mathematics  and  Science 

maker,  the  mechanic,  and  the  promoter.  Each  one  has  applied 
his  knowledge  to  his  labor,  and  the  finished  product  is  the  re- 
sult of  the  accumulated  researches  and  experiences  of  these 
men  in  turn.  Imagine  the  invention  of  the  steam  engine, 
the  sewing  machine,  the  typewriter,  the  adding  machine,  the 
lathe,  the  printing  press,  the  automobile,  the  airplane,  with- 
out an  expert  mathematical  training  on  the  part  of  the 
inventors. 

The  recent  war  with  its  wonderful  though  terrible  inventions, 
which  sprang  into  being  from  all  sides,  illustrates  most  forcibly 
the  value  of  mathematical  ability,  because  every  machine  from 
the  gas  mask  to  the  submarine  involved  the  exercise  of  mathe- 
matical genius.  From  the  moment  of  its  first  conception  in 
the  mind  of  the  inventor  to  the  actual  firing  of  the  first  shot 
on  the  battlefield,  the  giant  field  gun  is  a  product  of  mathe- 
matics. The  battle  of  Messines  Ridge  was  won  by  engineers 
who  skillfully  tunneled  the  hills.  Dr.  Nichols,  of  the  Uni- 
versity of  Virginia,  contributed  to  our  wonderful  shipbuilding 
exploit  by  successfully  solving  an  equation  to  the  ninth  degree.1 

Moritz  shows  our  commercial  dependence  on  mathematics. 
Thales,  the  ancient  mathematician,  was  by  profession  a  mer- 
chant, and  yet  he  studied  mathematics  for  the  intrinsic  value 
it  held  for  him.  Such  modern  business  problems  as  equation 
of  payments,  theory  of  interest,  valuation  of  debenture  bonds, 
amortization  of  interest-bearing  notes,  life  insurance  mor- 
tality tables,  distribution  of  dividends,  casualty  insurance, 
and  a  thousand  others,  have  their  solution  through  the  ap- 
plication of  pure  mathematics.  Statistical  work,  which  is 
used  in  hundreds  of  transactions  and  enterprises  of  every 
kind,  is  all  mathematics.  Business  executives  say  they  prefer 
1  C.  E.  White  in  School  Science  and  Mathematics,  January,  1919. 


Divisions  of  Elementary  Algebra  31 

mathematically  trained  men  because  they  are  more  methodical, 
exact,  and  resourceful,  and,  therefore,  efficient.1 

Algebra  Has  an  Important  Position.  From  a  strictly 
utilitarian  standpoint,  algebra  has  a  firm  claim  for  an  im- 
portant position  in  our  program  of  studies.  It  offers  the  only 
means  of  solving  many  of  the  problems  connected  with 
engineering,  architecture,  navigation,  surveying,  meteorology, 
geology,  physiology,  and  even  psychology,  if  we  accept  the 
Weber-Fechner  law,2  astronomy,  physics,  chemistry,  and 
many  other  branches.  No  mechanic  or  artisan  can  read 
intelligently  his  trade  journal,  a  technical  book,  an  article 
in  the  encyclopedia,  without  a  knowledge  of  the  universal 
language  of  algebra.  There  are  27,000  volumes  in  the  Naval 
Observatory  at  Washington  which  absolutely  require  a 
mathematical  training  for  their  perusal.3  Statistics  of  all 
kinds  are  given  in  equations  or  mathematical  formulas.  The 
graphical  treatment  is  employed  by  the  economist,  the  business 
expert,  the  physician,  the  dietitian,  and  men  of  hundreds  of 
other  professions.  The  handling  of  pig  iron  in  the  modern 
foundry  is  a  result  of  long  continued  analytical  experiments 
based  on  algebraic  formulas. 

Algebra  Is  Necessary  to  Secure  a  Higher  Education.  It  is 
indispensable  to  the  future  student  of  astronomy,  physics, 
chemistry,  and  higher  mathematics.  The  modern  engineering 
world  and  all  technical  schools  are  forever  closed  to  him  with- 
out a  mastery  of  this  subject.  Just  the  other  day  the  writer 
had  an  illustration  of  this.  A  young  man  who  had  been 

JR.  E.  Moritz,  "Our  Relation  of  Mathematics  to  Commerce";  School 
Science  and  Mathematics,  April,  1919. 

2  C.  E.  White  in  School  Science  and  Mathematics,  January,  1919. 
N    'Schultze, "  The  Teaching  of  Mathematics  " ;  The  Macmillan  Company,  1912. 


32      Supervised  Study  in  Mathematics  and  Science 

recently  graduated  from  high  school  found  that  it  was  possible 
for  him  to  go  to  college  but  he  had  been  allowed  to  go  through 
this  school  without  algebra  and  geometry,  and  he  found  to  his 
dismay  that  he  was  unable  to  matriculate  because  of  this 
deficiency  in  his  education.  Many  a  young  man  seems  for- 
ever doomed  to  a  nominal  wage  and  a  position  near  the  foot 
of  the  ladder  because  he  failed  to  fit  himself  for  higher  positions 
by  the  careful  study  of  mathematics. 

A  noted  inventor  once  said  that  he  had  no  use  for  algebra  01 
higher  mathematics,  but  in  the  next  breath  he  admitted  that 
he  hired  experts  to  work  out  for  him  all  problems  involving 
advanced  mathematics.  Mathematical  physics  and  mathe- 
matical chemistry  are  important  branches  of  science  to-day, 
and  every  phase  of  electrical  engineering  is  intimately  bound 
up  with  algebra  and  its  manipulations. 

The  writer  once  knew  a  man  who  lost  a  fine  position  as 
assistant  contractor  because  he  did  not  know  algebra,  al- 
though he  was  an  expert  workman.  His  would-be  employer 
felt  he  could  not  afford  to  have  a  man  in  this  responsible 
position  who  was  ignorant  of  this  subject,  as  occasions  some- 
times arose  which  demanded  a  knowledge  of  it. 

A  man  must  know  algebra  to  advance  in  the  automobile 
business.  A  modern  blue  print  with  its  mechanical  and 
mathematical  symbols  looks  like  a  Chinese  puzzle  to  the  lay- 
man, but  to  the  trained  mind  of  the  skilled  mechanic  it  is  as 
plain  as  the  printed  page. 

And  so  we  might  multiply  the  examples  of  the  practical 
value  of  algebra  indefinitely,  but  enough  has  been  said  to 
arouse  in  the  mind  of  the  pupil  the  strong  suggestion  that  here 
is  a  subject  which  has  a  bearing,  both  directly  and  indirectly, 
upon  his  future  as  well  as  upon  the  progress  of  the  world. 


Divisions  of  Elementary  Algebra  33 

Bird's-eye  View  of  the  Course.  —  A  few  minutes  might 
profitably,  be  spent  in  outlining  the  semester's  work.  Tell 
the  class  that  there  will  be  a  certain  amount  of  formal  work, 
similar  to  that  uone  in  arithmetic,  such  as  learning  how  to 
add,  subtract,  multiply,  and  divide  algebraic  terms  and  quan- 
tities, finding  factors,  reducing  fractions,  and  manipulating 
equations,  in  order  that  we  may  arrive  at  the  solution  of 
problems.  Explain  that  this  is  necessary  in  order  to  become 
familiar  with  the  tools  of  algebra,  just  as  the  carpenter  must 
learn  how  to  use  the  hammer,  the  saw,  the  square,  and  the 
compass,  before  he  can  build  a  house. 

Explain  that  from  time  to  time,  as  a  topic  is  finished,  we 
shall  have  a  sort  of  field  day,  when  we  may  exhibit  our  work, 
give  special  reports  of  certain  phases  of  the  subject,  and 
invite  our  friends  to  see  the  things  we  have  accomplished. 

Tell  the  pupils  also  that  we  shall  occasionally  have  contests, 
in  which  we  shall  choose  sides  to  see  which  side  can  win.  This 
may  be  done  with  fine  results,  as  will  be  illustrated  later,  upon 
the  completion  of  the  work  in  factoring. 

Again,  explain  that  during  the  year  the  class  will  learn  how 
to  solve  various  problems  which  will  be  drawn  from  business, 
agriculture,  the  vocations,  etc.  Tell  them  that  they  them- 
selves will  also  be  expected  to  make  up  and  solve  problems, 
and  that  data  drawn  from  current  events,  like  an  election  or 
a  ball  game,  will  be  used. 

Conditions  for  a  Successful  Preview. — Too  much  must  not 
be  attempted  in  this  first  lesson.  The  word  "inspirational" 
very  graphically  illustrates  the  underlying  motive  for  this 
lesson.  The  preceding  material  may  well  be  added  to  or  in 
part  eliminated,  as  seems  best  in  the  judgment  of  the  teacher 
himself.  It  is  given  simply  to  suggest  some  of  the  things  that 


34      Supervised  Study  in  Mathematics  and  Science 

may  make  this  first  meeting  of  the  class  an  inspiration  and  a 
forward  look. 

Illustrations  concerning  the  practical  value  will  have  more 
force  if  they  have  come  under  the  actual  observation  of  the 
teacher.  Indeed,  the  teacher  of  algebra  may  well  keep  a  note- 
book in  which  such  material  and  experiences  may  be  collected 
and  added  to  from  time  to  time.  Such  a  teacher  will  soon 
accumulate  a  valuable  set  of  illustrations,  and  one  that  will 
have  variety  and  modern  application.  The  historical  notes  will 
also  vary  with  the  teacher ;  he  must  ever  be  on  the  alert  for 
interesting  incidents  to  use  in  this  connection.  Such  material, 
aside  from  the  cases  which  come  under  his  own  observation, 
will  be  found  in  magazine  articles,  newspaper  articles,  and 
the  experiences  of  others  and  will  be  drawn  from  the  pupils 
themselves. 

LESSON  II 

UNIT  OF  INSTRUCTION  I.  —  INTRODUCTION 

LESSON  TYPE.  —  A  LESSON  IN  CORRELATION 

Program  or  Time  Schedule 

The  Review 5  minutes 

The  Assignment 30  minutes 

The  Study  of  the  Assignment 25  minutes 

Purpose.  —  The  purpose  of  this  lesson  is  threefold :  (i)  to 
gain  the  confidence  and  free  expression  of  the  pupil  by  getting 
him  to  do  something  he  knows  how  to  do ;  (2)  to  review  and 
reemphasize  some  of  the  fundamental  operations;  and  (3)  to 
link  together  arithmetic  and  algebra  by  showing  their  inter- 
relationship. 


Divisions  of  Elementary  Algebra  35 

The  Review.  —  Since  this  is  the  first  review  work  of  the 
year  and  there  has  been  no  assignment,  a  few  questions  like 
the  following  might  be  asked,  suggested  by  the  "  Inspirational 
Preview  " : 

How  did  we  get  the  word  algebra  ? 

Name  some  countries  which  have  contributed  to  the  devel- 
opment of  this  subject. 

For  what  is  Vieta  important  ? 

Relate  an  interesting  incident  connected  with  the  history  of 
this  subject. 

Name  some  professions  which  presuppose  a  knowledge  of  algebra. 

The  Assignment. 

1 .  Information  given  by  the  teacher  regarding  the  (a)  func- 
tion and  (b)  applications  of  algebra. 

2.  Review  of  the  fundamental  processes  in  arithmetic. 

3.  Recognition   of    the   interdependence   of    algebra   and 
arithmetic. 

The  Function  of  Algebra.  Sir  Isaac  Newton  called  algebra 
"  Universal  Arithmetic."  Comte  defined  arithmetic  as  the  sci- 
ence of  values  and  algebra  as  the  science  of  functions.  Alge- 
bra deals  historically  and  primarily  with  number.  Primeval 
man,  desiring  to  count  his  possessions,  used  various  forms  of 
tallies.  The  ancient  Roman  used  the  pebble,  setting  aside  one 
for  each  article  counted.  The  ten  fingers  or  digits  were  early 
used  in  counting,  thus  giving  us  the  term  used  in  numeration, 
and  later  evolving  into  the  ten  digit  or  decimal  (i.e.  decem,  ten) 
system.  Gradually  numbers  became  of  interest  because  they 
allowed  combinations.  We  therefore  use  symbols  to  stand  for 
or  represent  things,  later  substituting  the  thing  itself.  Thus 
we  are  enabled  to  derive  a  general  statement  which  may  be 
applied  to  all  similar  cases. 


36      Supervised  Study  in  Mathematics  and  Science 

We  have  done  something  of  this  in  our  arithmetic  when 
we  used  the  statement  BxR  equals  P.  Any  problem  in 
percentage  may  be  reduced  to  this  formula  or  some  variant  of 
it.  When  the  substitutions  are  made,  the  problem  may  be 
solved.  As  in  the  case  of  percentage  we  have  used  the  first 
letter  of  the  word  indicated,  so  in  algebra  we  also  represent  the 
unknown  by  letters.  But  in  that  case,  not  knowing  what  the 
words  may  be,  we  do  not  try  to  use  the  initial  letter.  It  has 
become  customary  to  employ  the  least  used  or  last  letters  of 
the  alphabet,  x,  y,  and  z. 

We  find,  therefore,  that  algebra  becomes  a  general  science 
while  arithmetic  remains  a  particular  science,  and,  though 
they  may  be  said  to  resemble  each  other  in  some  respects,  in 
reality  algebra  becomes  a  new  and  more  general  method  of 
manipulating  quantities. 

Applications  of  Algebra.  The  fundamental  processes  of 
addition,  subtraction,  multiplication,  and  division  of  whole 
numbers  and  fractions  form  the  foundation  stone  of  the 
formal  work  in  algebra  as  well  as  in  arithmetic,  but  we  shall 
find  that  their  functions  and  applications  in  this  new  science 
reach  a  breadth  which  is  impossible  in  arithmetic.  In  solv- 
ing problems  in  arithmetic  we  are  always  limited  to  things 
concrete,  but  in  algebra,  through  the  use  of  the  unknown,  we  are 
at  liberty  to  sweep  the  whole  field  in  our  solution.  It  is  for  this 
reason  that  algebra  assumes  a  universal  value  of  its  own. 

Review  of  the  Fundamental  Processes  in  Arithmetic.  —  A 
half  hour's  time  may  well  be  spent  in  reviewing  the  funda- 
mental operations  in  arithmetic  and  in  bringing  out  the  errors 
which  are  very  common  to  most  first  year  pupils,  such  as  the 
product  when  multiplying  a  number  by  unity  or  zero,  the 
manipulation  of  simple  fractions,  mixed  numbers,  reduction 


Divisions  of  Elementary  Algebra  37 

of  a  fraction  to  unity  in  cases  like  i  =  i,  cancellation,  etc.  It 
will  amaze  one  to  find  out  the  number  of  pupils  who,  sup- 
posedly ready  for  algebra,  cannot  do  correctly  some  of  the 
simplest  fundamental  operations  in  arithmetic.  It  will  sur- 
prise not  only  the  teacher  but  the  pupil  as  well,  since  he  is 
apt  to  feel  that,  having  passed  the  final  preacademic  examina- 
tion in  this  subject,  he  is  somehow  endowed  with  a  supreme 
and  unfailing  knowledge  of  arithmetic  for  all  time. 

The  teacher  may  apply  some  of  the  operations  of  arithmetic 
to  algebra,  but  it  is  best  not  to  do  too  much  of  this  at  first. 
For  instance,  after  the  fraction  £•  has  been  reduced  to  i,  we 
might  take  •*£•  and  show  that  it  reduces  to  ix,  or  after  a  review 
of  the  multiplication  of  fractions,  as  iXi  =  i,  we  might  show 
thatiXf  =  ir,  keeping  in  mind  that  we  are  not  at  present 
interested  so  much  in  the  teaching  of  algebra  as  we  are  in 
showing  that  our  present  knowledge  of  arithmetic  will  be  of 
constant  help  to  us  ;  and  in  emphasizing  that  we  are  not  going 
to  work  in  a  strange  land  but  will  have  with  us  old  friends. 

Interdependence  of  Algebra  and  Arithmetic.  The  nomen- 
clature is  similar  to  that  of  arithmetic  and  even  the  mechani- 
cal method  of  setting  down  the  problem,  as  in  division,  will 
be  the  same. 

We  have  already  used  the  simple  equation  in  the  lower 
grades  when  in  teaching  addition  we  give  such  problems  as 
"  two  and  what  make  five?  "  In  arithmetic  we  call  this  the 
Austrian  method,  but  in  algebra  it  becomes  finding  the  un- 
known. 

We  have  learned  that  we  cannot  add  together  5  apples  and 
3  oranges  except  to  say  that  we  have  5  apples  plus  3  oranges. 
So  in  algebra  we  cannot  add  $a  and  36  except  by  indicating 
50  plus  3&. 


38     Supervised  Study  in  Mathematics  and  Science 

There  is  an  intimate  relation  between  exercises  in  removing 
the  parentheses  in  arithmetic  and  algebra,  such  as  (18  —  2 
^4X2),  in  which  the  order  of  performing  the  operations  and 
rules  for  removing  the  signs  of  aggregation  are  identical. 

In  the  use  of  the  question  mark  to  indicate  what  is  desired, 
we  have  in  arithmetic  simply  anticipated  the  use  of  the  un- 
known x  in  algebra,  in  such  an  example  as  this : 

3     5     ? 
—  y  — 

8Q       Q 
o      o 

The  graph  is  used  in  many  exercises  in  arithmetic  to  show 
data  which  are  used  in  the  problem.  These  statistical  graphs 
are  a  very  important  phase  of  elementary  algebra.  (See 
Unit  of  Instruction  XI,  Chapter  Two.) 

We  have  already  referred  to  the  use  of  the  formula  in 
arithmetic.  In  addition  to  BxR  equals  P,  we  have  a  number 
of  others  relating  to  problems  in  interest,  as  P  equals 
7-7-($iXrXO-  Again,  in  measurement  we  have  A  equals 
&X/z;  circumference  of  a  circle,  or  O,  equals  trXD. 

Many  texts  in  arithmetic  also  to-day  have  a  section  covering 
simple  linear  equations  of  one  unknown  which  are  solved 
algebraically. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 

Ten  or  more  examples  to  illustrate  the  common  errors  made 
in  arithmetic  as  suggested  in  the  paragraph  on  fundamental 
processes. 

1.  48X0=  7.   i+i-+|  = 

2.  oXio£=  8.  4i-2i  = 

3-  4|X5iV=  9-    12-^4X3+6  = 

4-  m+*i=  10-   48-3i  = 
6.   f^|=                                  11. 

6.   fXiXf  = 


Divisions  of  Elementary  Algebra  39 

//  or  Average  Assignment. 

Eight  or  ten  examples  to  illustrate  the  similarity  of  arith- 
metic and  algebra  as  explained  in  the  paragraph  on  inter- 
dependence. 

12.  32  and  what  make  50? 

13.  17  and  (x)  make  25,  what  is  x? 

14.  23  and  32  give? 

16.   67  times  7  give  (x)  ? 

16.  7  books  plus  3  chairs  plus  2  books  plus  6  chairs  equal  (?) 

books  plus  ( ?)  chairs. 

17.  5&+3c  +  2&+ic  =  (?)  6  +  (?)c. 

18.  3#+7:r  =  how  many  x? 

19.  If  b  =  5  and  area  =  1 20,  find  the  altitude  in  formula,  area  = 

bXh. 

Ill  or  Maximum  Assignment. 

20.  Mention  some  other  similar  cases  of  algebraic  applications 
in  arithmetic. 

21.  Give  some  formulas  besides  those  stated,  which  you  have 
had  in  arithmetic. 

22.  Bring  to  class  some  illustration  of  the  use  of  the  graphic 
means  of  showing  statistics. 

23.  Report  on  the  origin  of  symbols  of  operation.     (Slaught 
and  Lennes'  "Elementary  Algebra,"  p.  7,  and  Hawkes,  Luby, 
and  Teuton's  "First  Course  in  Algebra,"  pp.  4,  5.) 

The  Silent  Study  Period.  —  As  soon  as  the  time  for  the 
study  period  arrives,  the  pupils  should  commence  work  upon 
the  lesson  assigned  for  the  next  day.  This  assignment  in 
three  sections  has  been  fully  explained  in  Chapter  One.  The 
complete  assignment  should  always  be  placed  upon  the  board 
before  the  class  assembles,  so  that  there  may  be  no  delay  in 
commencing  to  work. 


4<D      Supervised  Study  in  Mathematics  and  Science 

Since  the  exercises  above  suggested  are  simple,  the  pupils 
will  probably  not  have  much  trouble,  but  they  should  under- 
stand that,  in  case  anyone  finds  difficulty  with  the  work,  he 
has  the  privilege  and  is  expected  to  raise  his  hand  for  aid. 
The  teacher  may  then  step  quietly  to  his  desk  and  find  out 
what  the  difficulty  is. 

It  is  important  that  the  teacher  and  the  pupil  realize  that 
the  teacher  is  not  expected  to  do  the  pupil's  work  for  him. 
The  teacher's  part  is  to  direct  attention  to  his  difficulty.  The 
obstacle  should  be  skillfully  cleared  up  through  the  redirected 
effort  of  the  pupil  along  the  right  path. 

If  the  point  raised  seems  likely  to  be  a  stumblingblock  to 
others,  the  teacher  may  step  to  the  board  and,  calling  the 
attention  of  the  class  to  the  difficulty,  make  necessary  expla- 
nations which  will  at  once  be  a  help  to  all.  It  often  happens 
that  some  little  point  was  overlooked  in  the  explanation  before 
the  class,  which  may  now  be  made  clear. 

Encourage  the  pupils  to  do  as  much  of  the  assigned  lesson 
as  possible  during  the  study  period.  The  more  the  pupil  does 
in  the  classroom,  the  better  will  he  understand  his  work. 

Summary  on  the  Review.  • —  Each  day  should  provide  some 
definite  review  of  the  preceding  day's  work,  some  definite 
advance  in  the  mastery  of  the  subject,  and  some  definite  work 
assigned.  The  pupil  then  becomes  conscious  of  something 
positive  having  been  accomplished.  Nothing  is  more  detri- 
mental to  the  morale  of  the  pupil  than  for  him  to  feel  that  a 
day's  work  or  a  recitation  period  has  been  wasted  or,  at 
least,  has  passed  without  some  definite  advance.  When  our 
pupils  realize  that  nothing  will  be  allowed  to  interfere  with  the 
day's  work,  we  shall  find  that  they  will  be  more  anxious  to 
eliminate  their  absences.  Friday  or  the  day  before  or  after 


Divisions  of  Elementary  Algebra  41 

a  vacation  or  some  circumstance  is  allowed  often  to  break 
up  the  routine  of  steady,  purposeful  work.  With  the  proper 
attitude  and  evaluation  of  each  day's  importance,  however,  the 
teacher  can  make  every  meeting  of  the  class,  no  matter  under 
what  disadvantages,  a  distinct  step  forward.  The  morale 
of  the  allied  army  was  at  high  pitch  until  the  Rhine  was 
reached,  and  then  it  became  a  matter  of  anxiety  to  the  com- 
manders, because  the  soldiers  felt  that  their  object  had  been 
attained  and  that,  with  no  further  advance  being  made,  they 
were  merely  marking  time. 

The  review  should  be  short,  snappy,  and  purposive.  It 
really  takes  the  place  of  the  old  recitation,  as  such,  and  since 
it  is  much  shorter  in  time,  there  must  be  intensive  work  done. 
It  should  be  a  re-view  of  the  previous  day's  work,  a  clearing 
house  for  all  the  difficulties  encountered  in  the  study  of  the 
assignment,  and  an  opportunity  for  the  pupils  to  view  from  a 
new  and  broader  angle  the  work  studied  the  preceding  day. 

Summary  on  the  Assignment.  —  This  is  the  portion  of  the 
period  devoted  to  the  explanation  of  the  new  lesson.  The 
teacher  should  do  most  of  the  board  work,  the  pupils  following 
the  operations  at  their  seats.  As  far  as  possible  the  work 
should  be  developed  through  the  pupils,  since  they  will  then 
become  active  participants  and  their  interest  peculiarly  acute. 
Except  on  special  occasions,  all  board  work  should  be  elimi- 
nated, so  far  as  the  pupils  are  concerned.  Much  time,  chalk, 
and  patience  are  lost  when  all  or  part  of  the  class  are  working 
at  the  board.  The  inequality  of  time  needed  by  various  pu- 
pils, and  the  ease  with  which  the  brighter  ones,  out  of  a  job 
temporarily,  turn  to  things  not  connected  with  the  subject, 
cause  dismay  to  the  teacher  and  an  undesirable  diversion 
for  the  rest  of  the  class. 


42      Supervised  Study  in  Mathematics  and  Science 

But  when  the  teacher  himself  develops  the  work  on  the 
board,  and  the  class  follows  the  operations  on  paper,  every 
pupil  is  of  necessity  alert  and  attentive,  because  he  may  be 
called  upon  at  any  time  for  some  point.  The  entire  class  is 
therefore  kept  up  to  a  high  pitch  of  intensive  work.  The 
blackboard  as  a  means  of  visualizing  a  demonstration  before 
the  whole  class  has  a  distinct  value,  but  as  a  common  working 
ground  it  is  open  to  criticism. 

Summary  on  the  Study  of  the  Assignment.  —  As  has  already 
been  stated,  the  various  assignments  should  be  placed  upon 
the  board  before  the  meeting  of  the  class.  Let  them  be 
definite,  concise.  In  the  case  of  references,  the  exact  title  and 
page  of  the  book  referred  to  should  be  given.  The  use  of  the 
assignment  sheets  has  been  fully  explained  in  Chapter  One 
and  need  not  be  repeated  here. 

Besides  giving  the  pupils  something  definite  to  do  at  once, 
this  first  assignment  of  work  will  enable  the  teacher  almost 
immediately  to  single  out  the  poorer  pupils  and  those  who  are 
likely  to  be  the  workers  of  maximum  ability.  Plans  must  be 
made  at  once  to  take  care  of  both  classes, —  something  to  which 
the  supervised  study  period  is  especially  well  adapted.  A  re- 
arrangement of  the  seating  of  the  class  will  be  made  after  a  few 
days,  as  bad  results  sometimes  come  from  the  premature  an- 
nouncement that  permanent  seats  have  been  assigned.  A 
workable  plan  is  to  put  the  slow  workers  on  one  side  of  the 
room,  the  rapid  workers  on  the  other  side,  and  the  rest  of  the 
class  between  them.  If  the  class  can  be  approximately  so 
divided  early  in  the  term,  no  especial  attention  will  be  called 
to  this  classification,  and  the  reasons  therefor  may  remain  a 
secret  with  the  teacher.  This  arrangement  will  make  it  much 
easier  to  reach  the  two  extremes  of  the  class  and  give  to  them 


Divisions  of  Elementary  Algebra  43 

special  aid.  It  will  also  be  a  distinct  help  in  administering  the 
details  of  board  work,  individual  instruction,  and  the  use  of 
supplementary  material. 

Summary  on  the  Silent  Study.  —  The  importance  of  getting 
to  work  at  once  without  loss  of  time  should  be  explained  in  the 
beginning  of  the  term.  The  pupil  should  be  made  to  feel  that 
every  minute  is  valuable,  and  that  waste  of  time  will  not  be 
countenanced. 

Various  devices  for  keeping  track  of  the  completion  of  the 
different  sections  of  the  assignment  will  be  given  in  succeeding 
lessons.  The  object  of  these  sectional  assignments  is  solely  to 
aid  the  teacher  in  developing  to  the  utmost  each  individual 
pupil  in  the  class.  This  section  of  the  supervised  study  period 
should  be  the  most  important  part  of  the  hour,  because  it  is 
here  that  the  pupil  comes  into  personal  contact  with  the  teacher 
and  receives  first  hand  the  kind  of  aid  he  needs.  At  the 
same  time  a  certain  amount  of  sympathetic  relationship  is 
developed  on  the  part  of  both  teacher  and  pupil. 

LESSON  III 

UNIT   OF  INSTRUCTION  I.  —  INTRODUCTION 

LESSON  TYPE.  —  A  How  TO  STUDY  LESSON 

Program  or  Time  Schedule 

The  Review 5  minutes 

The  Assignment 30  minutes 

The  Study  of  the  Assignment 25  minutes 

Purpose.  —  The  crux  of  the  supervised  or  directed  study 
period  consists  in  the  definite  directions  for  the  proper  study 
of  the  subject  in  question,  and  careful  explanations  of  just 
how  to  study  rather  than  what  to  study.  It  is,  therefore, 


44      Supervised  Study  in  Mathematics  and  Science 

valuable  during  the  course  and  especially  at  the  beginning, 
that  very  definite  rules  for  study  should  be  outlined  and  in- 
sisted upon. 

The  Review.  —  Subject  Matter.  A  summary  of  the  pre- 
vious day's  assignment. 

Method.  Call  for  questions  on  the  exercises  assigned.  Ask 
for  a  few  leading  facts  brought  out  by  this  assignment,  such  as : 

1.  What  was  Newton's  definition  of  algebra?  Comte's  definition 
of  arithmetic  ?  (  Show  their  pictures  to  the  class.) 

2.  How  did  we  get  the  word  decimal? 

3.  Mention  some  formulas  used  in  arithmetic. 

4.  What  makes  algebra  a  broader  subject  than  arithmetic? 

Give  a  few  examples  like  those  in  assignments  I  and  II. 
Call  on  someone  who  completed  the  maximum  assignment  of 
yesterday's  lesson  to  give  the  answers  to  the  questions  in  this 
assignment. 

The  Assignment.  —  i.   Methods  of  study. 

2.  The  system  of  supervised  study  explained  by  the  teacher. 

3.  The  technic  of  the  textbook  in  algebra. 

4.  Definite  instructions  in  how  to  study  algebra. 
Methods  of  Study.  —  (a)    External     conditions.      In    the 

first  place  a  correct  physical  environment  is  necessary.  One 
must  be  in  a  comfortable  seat,  with  good  light  coming  over  the 
left  shoulder,  breathing  fresh  air,  and  in  a  room  of  proper  tem- 
perature. The  pupil  must  have  the  necessary  tools,  as  paper, 
well-sharpened  pencil,  ruler,  textbook,  etc.  Next,  the  pupil 
must  put  himself  in  the  proper  attitude  toward  the  subject 
and  concentrate  his  mind  upon  his  work.  And  then,  with  a 
determination  and  expectation  to  succeed,  he  is  ready  to  com- 
mence his  work.  All  these  prerequisites  will  in  time  become 


Divisions  of  Elementary  Algebra  45 

automatic  if  the  teacher  frequently  calls  the  pupil's  attention 
to  them.  It  is  easier  to  talk  about  concentration  of  mind  than 
it  is  to  achieve  it,  but  the  pupil  should  be  carefully  shown  that 
the  better  control  one  has  over  his  ability  to  keep  his  mind 
from  wandering,  the  better  student  he  will  be  and  the  sooner 
will  he  be  able  to  master  the  task  in  hand.  It  takes  will  power 
and  practice,  but  the  teacher  cannot  emphasize  too  strongly 
the  inestimable  value  of  this  acquirement,  if  once  formed 
even  to  a  limited  degree.  As  McMurry  suggests,1  one  of  the 
best  methods  of  acquiring  concentration  is  through  the 
employment  of  time  tests,  which  require  undivided  attention. 
When  we  must  do  a  certain  thing  within  a  definite  time,  we 
concentrate  upon  it.  With  sufficient  practice,  this  may 
become  habituated. 

(b)  Technical  factors,  i.  The  first  technical  factor  in 
proper  studying  is  the  sensing  of  the  problem.  If  the  pupil 
simply  goes  at  his  work  with  a  view  of  covering  a  certain 
amount  of  prescribed  ground,  without  a  realization  of  the 
problem  involved,  his  study  will  degenerate  into  a  mechanical 
grind.  Every  assignment,  as  stated  before,  should  have  some 
definite  object  or  problem,  around  which  the  lesson  will  re- 
volve. In  the  first  lesson,  as  outlined  in  these  pages,  it  was 
"Is  algebra  practical?"  In  the  second  lesson  it  was  "Is 
algebra  entirely  new?"  and  to-day  it  is  "  What  is  the  best 
way  in  which  to  study  algebra?  " 

2.  The  second  factor,  after  the  recognition  of  the  motive 
of  the  lesson,  is  bringing  to  bear  upon  it  all  our  present  knowl- 
edge of  the  problem  and  then  supplementing  the  problem  from 
our  text  and  possibly  other  books  and  sources. 

3.  We  next  seek  to  master  the  supplementary  material  by 

1  F.  M.  McMurry,  "How  to  Study" ;  Houghton  Mifflin  Co.,  1909. 


46      Supervised  Study  in  Mathematics  and  Science 

constantly  referring  to  our  present  knowledge  and  associating 
our  new  facts  with  data  already  known.  This  may  mean  the 
employment  of  the  memory,  which,  if  rightly  used,  will  be- 
come a  means  and  not  an  end.  Too  much  of  our  studying 
resolves  itself  into  memorizing  alone,  and  it  then  becomes  a 
detriment.  But  as  Miss  Earhart  well  says,  "  Memory  must 
not  be  substituted  for  thought  but  be  based  on  thought."1 

There  are  three  kinds  of  memorizing:  purely  arbitrary 
memorizing,  memorizing  based  on  reasoning,  and  remember- 
ing the  sequence  rather  than  the  things  themselves. 

4.  The  fourth  and  last  factor  is  the  application  of  our  data 
and  material  in  the  solving  of  our  problems  or  in  making  our 
new  power  a  part  of  ourselves. 

The  Supervised  Study  Organization.  A  few  words  about 
the  supervised  study  period  may  now  be  given.  Each  day's 
work  will  be  divided  into  three  parts :  the  review,  the  assign- 
ment, and  study  of  the  assignment.  During  the  review,  the 
previous  day's  work  will  be  re-viewed  in  summarized  form  and 
any  difficulties  cleared  up.  This  will  usually  take  about 
15  minutes.  The  new  work  will  be  explained  during  the 
second  time  division,  i.e.  that  of  the  assignment.  This  will 
take  about  20  minutes  but  may  vary  in  amount.  The  study 
of  the  new  lesson  will  take  place  the  last  25  minutes,  and  dur- 
ing this  period  the  pupil  will  be  expected  to  do  as  much  of  the 
new  work  as  possible.  Explain  that  he  may  feel  free  to  raise 
his  hand  if  he  finds  need  of  help,  that  he  must  not  expect  to 
have  the  teacher  do  his  work,  but  only  redirect  him  to  find  his 
own  trouble,  or  lead  him  to  see  wherein  his  line  of  procedure 
is  erroneous.  The  study  period  is  not  an  opportunity  for 

xLida  B.  Earhart,  "Teaching  Children  to  Study";  Houghton  Mifflin  Co., 
1909. 


Divisions  of  Elementary  Algebra  47 

getting  someone  else  to  do  the  pupil's  task  but  an  opportunity 
for  getting  proper  directions  so  he  may  be  able  to  do  it 
himself. 

Explain  that  the  assignment  will  be  placed  daily  upon  the 
board,  that  it  will  be  in  three  parts,  and  that  each  pupil  will 
be  expected  to  do  the  first  two  parts  and  as  many  pupils  as 
possible  to  do  the  third.  The  assignments  should  be  so 
arranged  that  much  of  the  work  may  be  done  during  the  period 
itself.  The  exact  amount,  of  course,  will  depend  on  the  pupil. 
It  has  been  found  in  work  of  this  kind  that  pupils  are  proud 
to  excel  their  classmates.  There  are  only  the  rarest  instances 
of  pupils  deliberately  doing  only  the  minimum  assignment. 

The  Open  Book.  The  pupils  are  asked  to  open  the  text- 
book in  algebra  while  the  teacher  explains  the  structure. 
Comparison  is  made  between  title-page  and  the  cover.  A  few 
words  regarding  the  position  or  personality  of  the  author,  the 
name  of  the  publisher,  and  the  date  of  the  book's  publication 
may  arouse  some  interest  in  the  author  as  a  second  teacher 
of  the  class ;  for,  of  course,  the  author  of  a  textbook  is  to  be 
regarded  as  a  teacher. 

Have  a  pupil  read  the  preface  and  then  ask  him  a  few 
questions  which  will  bring  out  the  reason  for  such  an  intro- 
duction to  the  book. 

Turn  to  the  table  of  contents  and  note  the  various  topics  and 
subtopics  of  the  subject.  Compare  this  with  the  table  of 
contents  of  some  other  text  in  algebra.  Note  the  number  of 
pages  one  or  two  units  of  instruction  cover  in  your  book  and 
in  some  other  text.  If  any  topics  are  to  be  omitted  from 
study,  mention  the  fact. 

If  your  text  has  answers,  give  a  few  words  as  to  their  use 
and  abuse.  Teachers  differ  in  their  opinions  regarding  the 


48     Supervised  Study  in  Mathematics  and  Science 

value  of  the  printed  answers,  and,  if  it  is  impossible  for  the 
teacher  to  train  the  class  to  make  them  a  side  issue  and  not 
the  most  worn  portion  of  the  book,  it  is  clear  that  their  publi- 
cation is  a  serious  mistake. 

Now  open  the  book  at  the  first  page,  calling  attention  at  the 
same  time  to  the  fact  that  the  paging  of  the  book  proper 
commences  at  this  point.  Announce  that  the  work  for  the 
next  day  will  begin  here. 

This  review  of  the  make-up  of  the  book  proper  may  seem 
irrelevant  to  the  study  of  algebra  and  a  waste  of  time,  but 
aside  from  the  value  of  the  general  knowledge  thus  gleaned, 
it  serves  as  an  introduction  to  the  text  with  which  the  pupils 
are  to  have  intimate  acquaintance  during  the  year.  It  is 
well  that  they  know  something  of  the  nature  of  the  tool  with 
which  they  are  going  to  work.  It  often  happens  that  things 
learned  incidentally  in  connection  with  a  subject  will  be  of 
greater  educational  value  than  the  subject  matter  itself.  After 
all,  our  children  are  coming  to  school  primarily  to  be  educated 
and  secondarily  to  learn  algebra,  Latin,  or  any  other  partic- 
ular subject.  It  is  through  these  subjects  that  we  hope  to 
attain  the  ends  of  education.1 

Instructions  in  How  to  Study.  A  few  mimeographed 
directions  may  now  be  distributed  to  the  class,  and,  after 
necessary  explanations,  the  pupils  may  be  told  to  insert  them 
in  their  books  for  future  reference.  Explain  that  you  may 
add  other  directions  from  time  to  time  as  the  class  progresses, 
and  suggest  that  each  pupil  should  feel  free  to  make  any 
suggestions  for  the  enlargement  of  the  list. 

1 "  The  Textbook— How  to  Use  and  Judge  It"  by  Hall-Quest,  The  Macmillan 
Company,  1918,  gives  a  full  discussion  of  what  might  well  be  covered  in  teach- 
ing pupils  how  to  learn  to  use  academic  tools. 


Divisions  of  Elementary  Algebra  49 

The  list  which  follows  is  by  no  means  perfect  or  complete ; 
it  is  simply  suggestive : 

SUGGESTIONS  FOR  EFFECTIVE  STUDYING 

Be  sure  you  understand  the  assignment. 

Study  the  meaning  of  the  type  of  problem  you  are  to  solve,  as 
suggested  in  its  name,  i.e.  highest  common  factor,  addition  of 
radicals,  etc. 

Recall  the  teacher's  explanation  of  the  new  work. 

Study  again  the  type  form  or  example  of  the  new  problems. 

Understand  thoroughly  what  is  wanted  before  you  begin  to  use 
your  pencil. 

Avoid  guesswork. 

Take  tune  to  think.  Do  not  rush  into  an  exercise  trusting  to 
luck  you  will  strike  it  right.  Be  sure  you  are  right ;  then  go  ahead. 

Be  sure  you  set  the  exercise  down  correctly  on  your  paper. 

Work  carefully ;  it  is  easier  to  avoid  mistakes  than  it  is  to  find 
them. 

When  you  find  a  new  application,  study  it  until  you  master  it. 
Expect  each  new  problem  to  be  different  from  the  one  preceding ; 
else,  we  would  never  advance. 

In  case  you  cannot  proceed,  raise  your  hand.  Do  not  expect 
the  teacher  to  find  your  mistake  but  to  direct  you  to  findlt  yourself. 

Be  neat  in  your  work.  A  good  workman  is  known  by  his  neat 
performance. 

Slovenly  habits  of  work  lead  to  slovenly  habits  of  thought. 

The  Study  of  the  Assignment.  —  The  assignment  for  to- 
morrow will  be  in  one  section  only.  It  will  consist  of  some 
questions  on  the  points  of  to-day's  lesson,  in  regard  to  the 
attitude  of  study,  factors  of  study,  the  technic  of  the  text- 
book and  the  list  of  directions  on  how  to  study.  These 
questions  will  help  to  focus  the  study  on  the  essential  features 
and  to  prevent  wrong  conclusions.  A  few  sample  questions 
are  given : 

What  is  the  proper  temperature  for  a  living  room? 
Why  should  the  light  come  over  the  left  shoulder  ? 


50     Supervised  Study  in  Mathematics  and  Science 

Suggest  a  good  method  of  practice  to  attain  concentration  of 
mind. 

What  was  the  problem  of  to-day's  lesson  in  biology? 

Which  of  the  three  kinds  of  memorizing  do  you  use  in  relating 
the  incidents  of  a  ball  game  ? 

Name  some  sources  of  supplementary  material  aside  from  the 
textbook. 

Is  the  preface  necessary  in  every  book  ? 

Which  do  you  think  the  author  compiled  first — the  table  of 
contents  or  the  index  ?  Give  your  reasons. 

Suggest  any  other  directions  than  those  given  to  you  on  the 
printed  list. 

Which  one  of  those  given  do  you  think  would  save  you  the  most 
work,  if  carefully  carried  out  ? 

BRING  PAPER  AND  PENCIL   TO-MORROW 

LESSON  IV 

UNIT   OF  INSTRUCTION   I.  —  INTRODUCTION 

LESSON  TYPE.  —  AN  INDUCTIVE  AND  How  TO  STUDY 

LESSON 

Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.  The  questions  on  the 
previous  lesson. 

Method.  Write  the  various  questions  assigned  yesterday 
upon  slips  of  paper  and  have  these  in  a  loose  pile,  face  down, 
upon  the  teacher's  desk.  Call  on  some  pupil  to  come  to  the 
front  of  the  room,  to  pick  up  one  of  the  slips  at  random  and, 
after  reading  it  aloud,  to  proceed  to  answer  it.  If  the  class 
accepts  this  answer  as  correct  and  complete,  ask  someone 
else  to  repeat  the  process,  and  so  on  until  all  the  questions  have 


Divisions  of  Elementary  Algebra  51 

been  answered.  If  any  question  is  not  answered  acceptably, 
replace  the  slip  in  the  pile.  This  method  of  review  will  further 
help  to  break  down  the  barrier  between  pupil  and  teacher,  to 
accustom  the  pupils  to  talking  before  the  class,  to  teach 
clearness  and  accuracy  of  expression,  and  to  test  the  judgment 
of  the  value  of  the  answer,  thus  giving  all  something  to  do. 
Furthermore  it  helps  to  review  thoroughly  the  essential  points 
of  the  preceding  day's  lesson. 

NOTE.  — The  lessons  in  algebra  will  not  be  based  upon  any  special  textbook, 
but  the  directions  will  be  found  applicable  to  any  textbook  upon  the  market. 
Copies  of  all  the  modern  texts  are  upon  the  teacher's  desk,  and  constant  refer- 
ence is  made  to  these  either  for  supplementary  examples  or  other  material. 
Inasmuch  as  the  present  writer  is  interested  chiefly  in  presenting  a  variety  of 
schemes  for  teaching  and  training  pupils  in  economical  and  effective  methods 
of  study,  it  is  hoped  that  the  point  of  view  herein  developed  will  be  compre- 
hensive enough  to  include  the  situations  that  may  arise  in  the  use  of  any  text  in 
algebra. 

The  Assignment.  —  i.  Instructions  in  how  to  study  the 
printed  page. 

2.  Treatment  of  illustrative  material. 

3.  Treatment  of  class  exercises  (a)  oral,  (6)  written. 
Instructions  in  How  to  Study.     The  pupils  will  open   their 

texts  and  have  their  attention  directed  to  the  "  definitions." 
It  will  be  noted  that  this  is  the  first  unit  of  instruction  as 
listed  in  the  table  of  contents.  (In  the  divisions  of  algebra 
outlined  in  Chapter  Two,  it  is  given  as  a  unit  of  recitation 
under  introduction;  authors  differ  in  the  arrangement  of  the 
material.) 

The  first  paragraph  is  read  carefully  and  the  central  or 
important  point  or  problem  discussed.  The  pupils  will  readily 
select  the  essential  point  of  this  paragraph.  If  there  are  any 
words  which  are  new  or  not  clearly  understood,  they  should 


52      Supervised  Study  in  Mathematics  and  Science 

be  immediately  defined  by  the  teacher.  Before  we  can  com- 
prehend the  sentence,  we  must  know  the  meaning  of  its  com- 
ponent parts.  Explain  the  use  of  italics  and  heavier  type. 
These  take  the  place  of  the  emphases  in  oral  speech.  These 
mechanical  means  call  the  attention  of  the  reader  to  the  im- 
portance of  the  word  or  phrase  and  should  be  specially  noted 
by  the  pupil. 

If  the  sentence  or  paragraph  is  not  clear  at  the  first  reading, 
reread  it  until  the  thought  is  mastered.  Insist  on  the  impor- 
tance of  making  reading  thought-producing,  and  not  simply 
a  mechanical  pronunciation  of  words.  The  language  of 
mathematics  is  absolute  and  therefore  cannot  be  read  rapidly 
or  slurringly ;  every  word  means  something. 

Now  have  someone  reproduce  the  paragraph  in  his  own 
words.  Call  upon  a  number  of  pupils  to  do  the  same  thing, 
thus  bringing  out  in  various  degrees  of  perfection  the  mean- 
ing of  the  assignment,  and  setting  up  a  little  rivalry  for  the 
best  work.  Emphasize  the  facts  that  we  know  what  we 
can  reproduce  in  our  own  words  and  that,  when  reproduced 
word  for  word  like  the  text,  we  are  thinking  more  of  the 
mechanical  reproduction  than  we  are  of  the  thought  to  be 
reproduced. 

The  above  outlined  study  of  the  paragraph  might  well  be 
applied  to  the  printed  page  of  any  book  which  is  a  subject  of 
study,  although  some  authorities  strongly  advise  that  the  first 
reading  of  the  page  or  section  be  made  as  a  whole  in  order  to 
get  the  general  sense  of  the  material.  In  algebra,  however, 
since  the  textual  matter  is  localized  in  its  meaning,  the  pre- 
reading  might  be  dispensed  with.  The  reader  will  note  also 
that  the  first  three  steps,  mentioned  in  the  preceding  lesson 
as  the  order  in  which  a  subject  should  be  studied,  have  been 


Divisions  of  Elementary  Algebra  53 

followed,  i.e.  the  point  of  view  or  problem,  the  data  or  mate- 
rial, and  its  mastery.  Its  application,  as  is  often  the  case, 
will  be  made  later.  It  often  happens  that  we  accumulate 
material  through  these  steps  for  some  length  of  time  before 
we  finally  bring  it  together  in  the  fourth  or  concluding  step. 

Treatment  of  Illustrative  Material.  When  we  come  to 
illustrative  material,  the  example  should  be  reworked  on 
paper.  Otherwise  the  pupil  will  mechanically  read  the  oper- 
ation, think  he  understands  it,  and  in  a  short  time  find  that 
it  has  slipped  away  from  his  consciousness.  This  reworking 
of  the  example  on  paper  will  also  help  to  fix  it  firmly  in  his 
mind  and  to  establish  each  step  thoroughly  as  it  is  written 
down,  provided  always  that  the  pupil  does  the  work  with  the 
motive  of  understanding  the  operations  as  they  are  evolved. 
Since  this  is  an  illustrative  or  model  lesson,  the  teacher  will 
also  do  the  work  which  he  will  expect  the  pupils  to  do  for 
themselves  in  their  future  study. 

For  instance,  suppose  this  formula  is  given : 

a  =  bXh. 

The  meaning  of  the  symbols  is  studied  and  then  the  value  of 
the  letters  in  a  specific  case  is  given  and  put  upon  the  board. 

Thus,  a  equals  20 ; 

h  equals  5. 

The  question  is,  what  is  the  value  of  b?  The  substitutions 
are  now  made  in  this  formula  and  the  solution  performed. 
This  operation  should  be  repeated  a  number  of  times  with 
varying  values  for  the  letters. 

Two  things  are  being  done  in  this  operation :  the  pupil  is 
learning  how  to  interpret  the  printed  word  by  his  clarified 
perception,  and  he  is  also  learning  the  fundamental  character- 


54      Supervised  Study  in  Mathematics  and  Science 

istics  of  algebra,  the  broad  application  of  the  algebraic  func- 
tion. It  might  be  well  at  this  point  to  request  some  pupil  to 
turn  to  the  introduction  of  Milne's  Standard  Algebra1  and 
read  what  that  author  has  to  say  concerning  it : 

The  basis  of  algebra  is  found  in  arithmetic.  Both  arithmetic 
and  algebra  treat  of  number,  and  the  student  will  find  in  algebra 
many  things  that  were  familiar  to  him  in  arithmetic.  In  fact,  there 
is  no  clear  line  of  demarcation  between  arithmetic  and  algebra. 
The  fundamental  principles  of  each  are  identical,  but  in  algebra 
their  application  is  broader  than  it  is  in  arithmetic. 

The  very  attempt  to  make  these  principles  universal  leads  to 
new  kinds  of  numbers,  and  while  the  signs,  symbols  and  definitions 
that  are  given  in  arithmetic  appear  in  algebra,  with  their  arithmet- 
ical meanings,  yet  in  some  instances  they  take  on  additional  mean- 
ings. .  .  . 

In  short,  algebra  affords  a  more  general  discussion  of  number  and 
its  laws  than  is  found  in  arithmetic. 

Since  with  this  introduction  the  pupil  has  an  idea  of  the 
manner  in  which  he  should  study,  the  teacher  should  further 
encourage  him  to  proceed  alone  in  his  study.  Questions  need 
to  be  asked  from  time  to  time,  however,  to  make  sure  that  the 
pupil  is  following  the  directions  and  getting  the  right  ideas. 
To  illustrate,  after  the  class  has  studied  some  paragraph  or 
section,  ample  time  having  been  allowed  for  the  use  of  the  dic- 
tionary, etc.,  ask  some  questions  about  it,  and  then  call  upon 
someone  to  state  the  problem  involved ;  someone  else  to  restate 
it  in  his  own  words ;  and  others  to  supplement  it  from  their 
own  knowledge  if  possible.  Thus  we  more  and  more  throw 
the  pupil  upon  his  own  resources  but  always  with  the  proper 
methods  of  procedure  before  him,  and  careful  supervision  on 
the  part  of  the  teacher  to  see  that  he  gets  the  correct  interpreta- 
tion. He  will  eventually  acquire  the  habit  of  study  as  outlined 

1  American  Book  Co.,  1914. 


Divisions  of  Elementary  Algebra  55 

above,  which  may  be  of  more  lasting  value  to  him  than  the 
algebra  itself. 

Treatment  of  Class  Exercises.  The  treatment  of  exercises  to 
be  worked  in  class  will  differ  somewhat  from  that  of  illustrative 
material.  Suppose  we  wish  to  take  up  such  exercises  as  the 
following : 

Read  and  explain : 

1.  a+b.  3.    aXb. 

2.  a— b.  4.   a+b. 

We  have  now  accumulated  our  material  and  are  ready  for 
its  application,  or  the  fourth  step.  Here  are  definite  examples 
of  what  we  have  been  studying  about  up  to  this  point.  All 
study  is  for  an  end.  As  the  final  end  of  algebra  is  the  solution 
of  problems,  so  an  intermediate  step  in  the  attainment  of  this 
end  is  the  ability  to  perform  the  mechanical  processes  which 
will  later  be  involved  in  their  solution. 

Class  exercises  will  therefore  be  of  two  kinds,  (a)  oral  and 
(6)  written. 

(a)  Oral  exercises.  Some  pupil  is  told  to  rise  and  read  the 
first  example.  He  is  then  asked  to  analyze  or  tell  the  meaning 
of  it,  which  should  be  something  like  this :  Two  general  mem- 
bers of  different  values  are  added  together  by  indication, 
a  added  to  b.  The  teacher  should  insist  on  complete  answers, 
told  in  technical  terms  and  in  simple  English.  Clear  thinking 
and  clear  expression  will  thus  be  unconsciously  habituated 
by  the  pupil. 

(&)  Written  exercises.  Exercises  like  the  following,  how- 
ever, may  preferably  be  treated  in  a  wholly  different  manner. 
Some  such  procedure  as  outlined  here  may  be  used  or  some 
modification  of  it : 


56      Supervised  Study  in  Mathematics  and  Science 
If  a =3,  b=  2,  and  c=%,  find  the  value  of  each  of  the  following : 

1.   ~  4-  3*°. 

0 

o      2«2  K     a~b 

2-  T  «+* 

b 

Most  of  the  board  work  should  be  done  by  the  teacher 
himself.  The  pupils  should  remain  at  their  seats  and  either 
work  on  paper  or  tell  the  teacher  what  to  write  upon  the 
board.  This  elimination  of  board  work  by  the  pupils  will 
result  in  a  more  efficient  use  of  the  time  of  the  period,  as  all 
members  of  the  class  will  be  either  at  work  or  on  the  lookout 
for  possible  questions.  Every  mark  put  upon  the  board  should 
first  be  supplied  by  some  member  of  the  class  and  accepted  by 
all  as  correct.  Thus  each  member  becomes  personally  inter- 
ested in  the  operations  and  alert  to  give  directions  or  to  detect 
errors.  The  class  is  thus  kept  up  to  a  high  tension  and  inten- 
sive work  may  be  accomplished.  The  board  work  becomes 
a  check  and  not  a  key,  and  the  pupils  feel  that  they  have  had 
a  real  part  in  its  development. 

To  illustrate,  the  work  on  the  first  example  given  above 
will  proceed  like  this:  the  teacher  will  ask  someone  to  read 
the  example  and  to  explain  how  to  make  the  substitutions.  He 
will  then  write  it  upon  the  board,  directing  the  pupils  to  do  like- 
wise on  their  papers.  Another  pupil  will  then  be  called  upon 
to  tell  what  is  to  be  done  next.  As  the  pupil  states  the  various 
steps,  the  teacher  will  place  them  upon  the  board,  the  class 
meanwhile  doing  the  same  on  their  papers.  The  pupil  reciting 
will  say  something  like  this  :  The  expression,  6a,  means  that 
the  literal  number  a  is  taken  6  times,  or  that  a  multiplied  by 


Divisions  of  Elementary  Algebra  57 

6  constitutes  the  term  in  the  numerator,  and  that  the  product 
is  to  be  divided  by  b.  Unless  we  give  these  literal  numbers 
some  values,  the  actual  division  can  only  be  indicated  as  in 
the  example.  But  if  we  assign  some  arbitrary  values  to  the 
literals,  we  may  substitute  these  values  in  the  expression, 
perform  the  necessary  operations,  and  reduce  to  its  simplest 
term.  In  this  case,  since  a  is  given  the  value  3  and  b  the  value 
2,  we  find  that  6a  is  equivalent  to  18,  and  this  divided  by  2,  or 
the  value  of  b,  gives  us  the  result,  or  9.  The  work  on  the 
blackboard  will  appear  as  follows : 

6a     6X3     18 

— -  = -  =  —  =  9.   Ans. 

022 

In  this  way,  it  would  be  well  for  the  teacher  to  work  on  the 
board,  with  the  assistance  of  the  pupils,  these  six  examples,  in 
order  that  the  pupils  may  learn  how  to  handle  written  exer- 
cises. When  written  work  is  next  required,  it  will  be  safe  to 
assume  that  they  will  know  how  to  go  about  their  work,  after 
one  or  two  typical  demonstrations  have  been  given  by  the 
teacher.  Work  of  this  kind  is  oral  or  cooperative  studying 
and  should  characterize  every  general  assignment. 

The  Study  of  the  Assignment.  —  Assuming  a  list  of  30 
graded  exercises  in  the  textbook  in  use,  assign  as 

I  or  Minimum  Assignment.     Exercises  1-20. 

//  or  Average  Assignment.     Exercises  21-30. 

Ill  or  Maximum  Assignment.  Exercises  15-20  on  page  46 
of  Ford  and  Ammerman's  First  Course  in  Algebra,1  or  exercises 
on  page  7  of  Slaught  and  Lennes'  Elementary  Algebra.2 

Value  of  Outside  Work.  Many  of  the  introductory  lessons 
will  be  along  the  line  of  the  foregoing,  the  majority  of  the 

1  The  Macmillan  Company.       *  Allyn  and  Bacon. 


58      Supervised  Study  in  Mathematics  and  Science 

exercises  being  worked  in  the  class  under  the  supervision  of 
the  teacher.  Each  day  a  short  assignment  should  be  made, 
based  on  the  ground  covered  and  preferably  taken  from  outside 
texts,  especially  the  maximum  assignment.  This  ought  to  be 
in  the  nature  of  a  review  of  the  work  done  in  class  and  should 
be  short  enough  to  allow  the  majority  of  the  class  to  complete 
all  three  assignments.  These  examples  may  be  handed  in  the 
next  day,  but  the  best  way  for  the  teacher  to  make  sure  that 
the  principles  are  thoroughly  understood  is  to  work  out  on  the 
board  through  the  minimum  workers,  or  those  who  only 
completed  the  minimum  assignment  during  the  study  period, 
a  few  typical  examples  during  the  review. 

The  pupil  must  realize  that  the  teacher  is  interested  not  so 
much  in  what  the  pupil  has  done  as  in  what  he  can  do  now. 
//  pupils  could  be  made  to  know  that  work  done  outside  of  class 
is  important  only  in  so  far  as  it  makes  them  capable  of  doing 
something  the  next  day  in  class,  the  incentive  for  getting  other 
people  to  do  their  work  would  be  greatly  diminished.  The  out- 
side work  must  be  insisted  upon,  unless  the  periods  are  long 
enough  to  have  all  the  work  done  in  class,  but  the  credit 
should  always  be  allowed  largely  upon  the  ability  to  do  similar 
work  again  in  class.  It  is  the  same  rule  that  applies  through 
the  walks  of  life.  The  stenographer,  the  carpenter,  the  printer, 
the  dentist,  the  worker  of  every  sort  is  not  paid  for  the  record 
he  has  made  in  speed  or  the  house  he  has  built  or  the  books 
he  has  printed  or  the  bridge  work  he  has  done,  but  for  his 
ability  to  do  similar  work  again.  To  be  sure,  the  experience  has 
made  him  proficient,  but  we  pay  for  results  and  not  for  the 
practice  that  has  made  the  results  possible.  The  one  is 
indispensable,  but  the  other  is  the  criterion  by  which  all  of  us 
are  judged. 


Divisions  of  Elementary  Algebra  59 

The  next  two  or  three  lessons  in  the  textbook  may  be 
worked  out  in  a  manner  similar  to  the  above.  The  amount 
of  time  spent  on  the  work  will  of  course  depend  on  the 
book  used  and  on  the  teacher.  He  may  condense  it  into 
a  shorter  period  or  take  even  longer.  The  material  given  is 
merely  suggestive  and  no  teacher  is  expected  to  follow  it 
verbatim.  In  fact,  such  a  procedure  would  probably  pre- 
determine the  failure  of  supervised  study,  because  more  than 
anything  else  its  successful  operation  depends  on  the  origi- 
nality and  individuality  of  the  teacher  himself.  No  system  has 
been  or  ever  will  be  evolved  which  automatically  may  be 
operated  by  someone  and  without  change  or  adaptation  be  a 
success  for  everyone  else.  All  that  may  be  hoped  for  any 
method  is  that  it  be  suggestive ;  its  final  application  and  adap- 
tation, in  the  last  analysis,  lies  with  the  teacher  himself.  In 
the  words  of  Miss  Simpson,  author  of  a  companion  book  in 
this  series,  "  it  is  imperative  that  teachers  adapt  rather  than 
adopt  the  methods  suggested  in  these  lessons."  l 

LESSON  V 

UNIT   OF   INSTRUCTION  m.  —  ADDITION 

LESSON  TYPE.  —  AN  INDUCTIVE  LESSON 
Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Upon  the  completion  of  each  unit  of  in- 
struction, it  is  advisable  to  re-view  the  unit  in  toto.    This  may 
take  the  form  of  an  oral  or  written  review.    Various  methods 
111  Supervised  Study  in  History  ";  The  Macmillan  Company. 


60      Supervised  Study  in  Mathematics  and  Science 

should  be  used.  In  reviewing  Positive  and  Negative  Numbers, 
the  following  is  suggested : 

Method.  Write  upon  the  board  a  large  number  of  exercises 
covering  the  various  phases  of  this  topic,  and  call  on  different 
members  of  the  class  for  the  answers.  These  should  be  writ- 
ten upon  the  board  in  the  proper  place.  Send  a  pupil  to  the 
board  and  as  the  answers  are  given  have  him  write  them  down 
if  he  considers  them  correct.  If  he  calls  one  correct  when  it  is 
wrong,  he  must  take  his  seat  and  another  be  sent  to  the  board 
in  his  place.  Thus  the  pupils  are  tested  on  their  ability  to 
solve  the  problems  and,  also,  on  their  ability  to  judge  correct 
results.  It  might  be  well  to  select  someone  to  call  on  the  dif- 
ferent pupils  to  recite,  the  teacher  noting,  however,  that  all 
or  most  of  them  are  given  a  chance.  When  the  pupil  at  the 
board  makes  a  mistake,  then  the  leader  should  take  his  place, 
and  so  on.  Thus  the  review  becomes  socialized.  It  will 
provide  interest  for  work  that  too  often  is  needlessly  weari- 
some. 

The  Assignment.  —  i.  The  arithmetic  preview. 

2.  Recognition  of  the  problem. 

3.  Explanation  of  the  type  form. 

The  Arithmetic  Preview.  As  intimated  in  the  second  les- 
son (page  36),  the  pupils'  present  knowledge  of  arithmetic 
should  always  be  drawn  upon  when  possible  to  illustrate  the 
new  work.  A  few  examples  in  adding  numbers  are  followed  by 
the  implied  addition  of  literal  terms.  The  induction  should 
be  made  by  the  class  and  the  Commutative  Law  of  Addition 
deduced.  After  a  few  attempts,  a  workable  law  will  be 
developed  by  some  such  questioning  as  this :  how  much  is 
three  and  five  ?  five  and  three  ?  a  and  b  ?  b  and  a  ?  Ask 
whether  it  makes  any  difference  in  what  order  numbers  or 


Divisions  of  Elementary  Algebra  61 

letters  are  added.  If  the  answer  is  "  no,"  ask  someone  to 
state  this  principle  in  a  sentence.  Answer :  Numbers  can 
be  added  in  any  order.  Tell  the  class  that  this  is  a  law  of 
order  or  the  Commutative  Law. 

Then  broaden  this  principle  when  two  or  more  numbers  are 
grouped,  as  (5  plus  6)  plus  4.  What  is  the  sum?  (5  plus  4) 
plus  6?  The  sum  is  the  same.  Then  we  broaden  the  above 
law  to  include  groups.  Ask  someone  to  state  the  revised 
principle.  Answer :  Numbers  may  be  added  in  any  order  or 
group.  Tell  them  that  this  is  the  Associative  Law  of  Addition. 

Recognition  of  the  Problem.  Ask  the  pupils  what  is  meant 
by  a  term,  a  monomial.  The  above  illustrations  are  all 
monomials.  Therefore  the  first  problem  under  Addition  will 
be  addition  of  monomials,  which  becomes  our  first  problem. 

Explanation  of  the  Type  Form.  Place  these  examples  upon 
the  board : 

Add: 

1.  3  2.  3  boys  3.  30 

j_  5  b°ys  .5^ 

There  will  be  no  difficulty  with  the  first  two.  Ask  in  3,  what  a 
stands  for.  Someone  will  say  "  boys."  But  might  it  not 
stand  for  girls  or  houses  or  almost  anything?  The  class 
will  readily  see  that  a  may  stand  for  anything  and  therefore 
the  answer  to  the  example  will  be  8a. 

Repeat  with  other  simple  monomials,  all  positive.  Then 
put  these  examples  on  the  board : 

Add: 

1.  5  2.  5  dollars  3.  50 

—  2  —2  dollars  —  20, 

With  their  previous  knowledge  of  positive  and  negative 
numbers,  the  class  will  see  that  in  each  case  the  coefficient  is  3. 


62      Supervised  Study  in  Mathematics  and  Science 

ASSIGNMENT  AND  STUDY  SHEET 

SUBJECT    Elementary  Algebra  PERIOD    2d 

DATE    September  7,  1921 
UNIT  OF  INSTRUCTION      Addition  (III) 
UNIT  or  RECITATION      Addition  of  monomials  (I) 
UNIT  OF  STUDY      Examples  1-25,  text 
LESSON  TYPE      Inductive 


REVIEW: 
Positive  and  Negative  numbers.    Exer- 
cises from  Wheeler's  Examples  in  Algebra, 
pages  4-7. 

MEMORANDA 
Examples  on  board.     One  writes 
answers  which  others  give.    Change 
when    mistake    is    made.     Work 
out  laws. 

ASSIGNMENT  : 
i.  Arithmetic  preview. 
2.  Recognition  of  new  problems. 
3.   Explain  type  form. 

What  are  monomials? 
Add  :  3            3  boys 
5            S  boys 

30 

i.  MINIMUM 
Exercises  1-20 

In  text. 

2.  AVERAGE 
Exercises  21-25 

In  text. 

3.  MAXIMUM 
Exercises  43-50,  Wells  and  Hart's  New 
High  School  Algebra,  page  38. 

Involve     fractions     and     deci- 
mals. 

STUDY: 
See  that  the  signs  are  correctly  copied. 

Number  of  pupils  solving  minimum  assignment                    7 
Number  of  pupils  solving  average  assignment                     22 
Number  of  pupils  solving  maximum  assignment                  i 
Total                ~3o 
FIGURE  III 

Divisions  of  Elementary  Algebra  63 

Now  call  on  some  pupil  to  stand  and  solve  the  first  exam- 
ple, which  may  be  : 

Add  :     20. 


Similar  exercises  may  be  given  orally.  They  may  well  be 
supplemented  with  others  from  the  board  until  the  principle 
is  well  understood. 

•"  Then  a  typical  example  like  the  following  may  be  developed 
on  the  board  and  the  class  set  to  work  on  the  study  of  the  new 
assignment  : 

Add:     2X,  %x,  —x. 

The  Study  of  the  Assignment.  —  Assuming  the  textbook  in 
use  gives  a  list  of  25  similar  exercises,  make  the  following 
assignments  : 

/  or  Minimum  Assignment.     Exercises  1-20. 

//  or  Average  Assignment.    Exercises  21-25. 

777  or  Maximum  Assignment.  Exercises  43-50,  page  38, 
in  Wells  and  Hart's  New  High  School  Algebra.1  These 
exercises  are  similar  but  a  little  more  difficult,  involving 
fractions  and  decimals. 

The  Silent  Study  Period.  —  The  Assignment  Sheet.  The 
division  of  the  assignment  into  three  parts,  as  suggested  in 
Hall-Quest's  book  on  Supervised  Study,  has  been  fully  ex- 
plained in  Chapter  One  of  the  present  volume,  which  should  be 
reread.  The  assignment  numerals  only  should  be  used  when 
designating  the  sections  in  placing  the  assignment  upon  the 
board  and  this  should  always  be  done  before  the  class  as- 
sembles. A  sample  sheet,  made  out  to  conform  to  this  lesson, 
is  given  in  full  on  page  62.  It  will  aid  the  teacher  materially 

1  D.  C.  Heath  and  Co. 


64      Supervised  Study  in  Mathematics  and  Science 

if  he  will  make  these  sheets  out  conscientiously  during  the 
term.  There  will  then  be  no  confusion  or  waste  of  time  if 
additional  exercises  are  needed  during  the  class  period.  Many 
precious  moments  are  saved  by  a  little  foresight  and  planning. 
A  lack  of  prearranged  plans  may  also  break  down  the  morale 
of  the  class.  Pupils  are  quick  to  respond  to  fine  or  poor 
executive  ability  when  either  is  exhibited  by  the  teacher. 
Napoleon  was  one  of  the  world's  greatest  generals  because  he 
had  the  absolute  confidence  of  every  soldier  under  him. 
Teachers  are  generals  of  a  little  school  army  and  the  morale 
of  the  one  is  analogous  to  that  of  the  other. 

The  Completion  of  the  Assignment.  The  minimum  and 
average  assignment  should  cover  the  amount  of  work  that 
the  teacher  would  ordinarily  give  under  the  old  method  of  only 
one  assignment  for  all.  That  is,  the  ordinary  lesson  for  the 
next  day  would  be  about  fourteen  exercises  in  the  text.  But 
these  have  been  broken  up  into  two  sections,  the  first  of  which 
should  be  worked  by  all  within  the  25  minute  study  period ; 
if  not,  then  the  pupils  who  fail  to  complete  this  part  need 
special  attention. 

All  the  class  is  expected  to  have  completed  the  average 
assignment  before  the  next  day  and  some  pupils  will  do  so 
before  the  end  of  the  period.  Those  having  trouble  will 
have  their  work  taken  up  during  the  review. 

The  maximum  assignment  is  designed  primarily  for  the 
brighter  and  quicker  pupils,  those  who  are  capable  of  doing 
more  than  the  average  amount  of  work.  They  should  be 
given  an  opportunity  of  trying  more  difficult  applications  of 
the  day's  work.  Not  many  will  complete  this  part  and  it 
should  not  be  demanded  from  all ;  but  when  done,  some 
system  of  giving  extra  credit  should  be  used.  A  good  method 


Divisions  of  Elementary  Algebra  65 

is  to  add  half  a  credit  to  the  monthly  grade  for  every  day  that 
the  maximum  assignment  was  done  correctly.  In  case  a 
pupil  did  this  correctly  every  day  for  a  month,  it  would  only 
mean  ten  extra  credits,  which  might  raise  the  grade  from  80  to 
90.  Very  few  would  attain  this  maximum  advance  grade, 
however.  But  the  teacher  must  be  careful  not  to  give  too 
much  credit  to  this  advance  work  and  so  discourage  the  slower 
worker;  it  should  be  the  aim  always  of  the  teacher  to  en- 
courage each  one  to  do  his  best  all  the  time. 

The  Teacher's  Duty  during  the  Study  Period.  As  soon  as  the 
study  period  begins,  all  start  to  work  on  the  next  day's  assign- 
ment. The  rate  of  speed  will  soon  become  uneven.  Some 
will  experience  no  difficulty  and  will  advance  rapidly; 
others  will  be  in  trouble  at  the  outset.  For  the  latter,  the  up- 
raised hand  will  quickly  bring  the  teacher  with  aid.  Thus 
help  comes  when  it  is  needed  and  at  the  time  that  the 
correct  direction  or  word  of  helpful  explanation  will  do  the 
most  good.  The  teacher  must,  of  course,  ever  be  on  the  alert 
to  see  that  his  help  is  corrective  or  suggestive  and  not  simply 
a  crutch.  It  should  be  directive  and  not  simply  finding  the 
mistake  for  the  pupil.  The  teacher,  in  quietly  moving  among 
the  pupils,  will  note  many  wrong  methods  and  incorrect  habits 
of  work  which  he  can  tactfully  correct.  Many  small  but 
important  things,  such  as  legible  handwriting,  neatness,  care- 
ful arrangement,  accuracy  in  copying  the  example,  may  be 
brought  to  the  attention  of  the  child  at  the  time  that  he  is 
working.  It  is  a  case  of  striking  when  the  iron  is  hot. 

Occasionally  a  glaring  error  and  its  possible  results  may  be 
called  to  the  attention  of  the  class ;  for  instance,  the  danger 
of  mistaking  a  poorly  formed  6  for  a  o  if  the  loop  is  not  care- 
fully attached  to  the  bend  of  the  figure  below  the  top.  Draw 


66      Supervised  Study  in  Mathematics  and  Science 

the  attention  of  the  class  to  the  fact  that  this  little  error 
invalidates  the  whole  later  process.  This  leads  excellently  to 
an  explanation  of  the  value  of  carefully  checking  the  work  as 
one  proceeds.  If  each  step  is  carefully  gone  over  and  checked 
for  errors  of  omission  or  commission,  before  the  next  step  is 
taken,  valuable  time  may  be  saved.  It  is  easier  to  avoid  mis- 
takes than  it  is  to  find  them.  (See  rule,  page  49.) 

Verification.  —  A  minute  or  so  before  the  close  of  the  period, 
check  up  the  work  done  in  class  by  the  pupils.  Various  meth- 
ods may  be  employed,  two  or  three  of  which  will  here  be 
explained.  Others  will  readily  suggest  themselves  to  the 
teacher. 
i  First  Method.  Have  some  printed  slips  like  this  : 


Name. 


Class Period 

Examples  completed 

Assignment  completed 

Time  spent  outside  of  class  on  to-day's 
lesson 


Teacher's  check. 


FIGURE  IV 

Such  a  form  can  be  filled  out  by  the  pupil  in  a  minute's 
time ;  the  teacher  can  collect  them  and  the  next  day  file  them 
with  the  papers  handed  in.  It  can  then  be  noted  how  much 
of  the  lesson  was  done  in  class,  and  how  that  amount  cor- 
responds with  the  examples  done  outside  of  the  class  period. 
This  method  will  give  the  teacher  the  names  of  the  pupils, 


Divisions  of  Elementary  Algebra  67 

who  failed  to  complete  the  minimum  assignment  in  class,  and 
these  should  have  special  attention  the  next  day.  This  check- 
ing will  also  serve  to  tell  the  teacher  whether  his  assignments 
are  too  long,  too  short,  or  about  right.  If  the  class  does  not 
approximately  conform  to  the  percentages  mentioned  in 
Chapter  One,  page  16,  something  is  wrong  in  the  assignment 
and  an  analysis  of  the  situation  should  be  made.  Incidentally, 
this  plan  will  take  care  of  the  roll  call. 

Second  Method.  Have  the  pupils  hand  in  all  exercises  com- 
pleted at  the  end  of  the  period.  Then  the  data  may  be 
computed  by  the  teacher.  The  next  day  the  examples  worked 
outside  of  class  may  be  handed  in  and  filed  with  the  others, 
which  will  thus  constitute  the  completion  of  the  assignment. 

Third  Method.  Just  before  the  close  of  the  period,  call  on 
all  who  have  completed  the  minimum  assignment  to  stand  or 
raise  their  hands ;  the  teacher  can  either  take  down  the  names 
or  have  the  pupils  hand  in  their  names  on  slips  of  paper. 
Then  in  like  manner  the  names  of  those  who  have  done  the 
average  and  maximum  assignments  may  be  obtained. 

Fourth  Method.  Have  the  pupils  check  on  their  papers  the 
point  they  had  reached  when  the  period  terminated;  then 
when  these  papers  are  handed  in  next  day,  the  teacher  may 
compile  his  own  lists. 

LESSON   VI 

UNIT  OF  INSTRUCTION  m.  —  ADDITION 

LESSON  TYPE.  —  AN  INDUCTIVE  LESSON 
Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 


68     Supervised  Study  in  Mathematics  and  Science 

The  division  of  the  time  of  the  class  period,  as  stated  at  the 
beginning  of  each  lesson,  is  that  followed  in  the  Canton  High 
School,  Canton,  N.  Y.,  where  supervised  study  has  been  in 
operation  since  1915.  The  day  is  divided  into  five  periods 
of  one  hour  each.  Longer  periods,  which  would  allow  the 
pupil  to  do  all  his  studying  in  school,  would  be  ideal,  but  in 
many  schools  such  a  program  could  not  be  administered.  If 
possible,  the  time  schedule  should  be  amplified,  but  the  above 
proportion  of  time  seems  very  adaptable,  where  different 
arrangements  cannot  be  made. 

The  Review.  —  Subject  Matter.    Addition  of  monomials. 

Method.  The  object  of  to-day's  review  is  (a)  to  assist  those 
pupils  who  had  trouble  with  the  assignment  and  (b)  to  give 
additional  drill  in  this  work  to  those  who  had  no  special 
difficulty  but  who  were  unable  to  complete  the  work. 

Those  who  have  completed  the  threefold  assignment  and 
who  have  mastered  the  addition  of  monomials  shouldl  be 
allowed  to  proceed  at  once  with  the  advance  assignment  in 
polynomials.  This  will  serve  as  an  inducement  for  intensive 
work  and  will  encourage  the  brighter  pupils  to  work  harder 
and  to  solve  all  the  exercises  during  the  period  if  possible. 
This  number  will  be  small  if  the  assignments  have  been  care- 
fully planned.  Those  who  have  done  part  of  the  maximum 
assignment  should  be  instructed  to  complete  it. 

Now  that  we  have  the  more  advanced  pupils  working  on 
the  next  lesson,  or  on  the  more  difficult  examples  of  the  maxi- 
mum assignment,  we  can  turn  our  attention  to  those  who  did 
not  complete  the  minimum  assignment  or  who  had  more  or  less 
difficulty.  These  pupils  may  be  treated  in  different  ways.  If 
several  failed  on  the  same  problem,  they  may  be  sent  to  the 
board  to  work  on  it  under  the  supervision  of  the  teacher.  The 


Divisions  of  Elementary  Algebra  69 

teacher  can  then  watch  their  work  and  soon  note  the  trouble. 
As  soon  as  a  pupil  finishes  one,  he  should  commence  on  the 
next  with  which  he  experienced  difficulty,  and  so  on.  This 
method  is  not  advised,  however,  for  reasons  already  stated 
against  board  work. 

A  better  method  would  be  for  the  teacher  himself  to  work 
out  the  problem,  with  the  pupil  directing  him  what  to  do,  the 
others  meanwhile  following  the  process  at  their  seats. 

Pupils  should  never  be  sent  to  the  board,  however,  to  work, 
for  the  benefit  of  others,  examples  which  they  solved  them- 
selves. Board  work  has  been  inordinately  stressed  in  mathe- 
matics. When  a  pupil  can  do  a  thing,  he  should  not  be  asked 
to  do  it  again ;  it  is  his  ability  to  do  something  which  he  could 
not  do  before  which  will  make  him  advance.  On  the  other 
hand,  such  exercises,  written  out  on  the  board  and  afterwards 
read  for  the  benefit  of  those  who  could  not  do  them,  will  be  of 
practically  no  value  to  them.  Pupils  can  learn  best  by  doing 
the  work  for  themselves. 

As  soon  as  the  problems  giving  trouble  have  thus  been 
solved,  the  remainder  of  the  review  period  should  be  devoted  to 
working  additional  ones  of  like  nature,  until  this  difficulty  has 
been  mastered. 

If  there  are  still  those  who  do  not  seem  to  be  able  to  under- 
stand the  problem,  they  should  be  given  individual  attention 
during  the  remainder  of  the  period.  The  teacher  must  feel 
that  his  special  task  is  to  help  the  less  capable ;  children  have 
varying  degrees  of  ability  and  it  is  the  peculiar  province  of  the 
supervised  study  scheme  that  the  backward  ones  are  thus 
given  special  attention  and  brought  up  to  the  standard  of  the 
class  as  quickly  as  possible.  The  old  process  of  the  elimination 
of  the  dull  pupil  must  give  way  to  the  new  idea  of  reaching  him 


yo     Supervised  Study  in  Mathematics  and  Science 

through  a  study  of  his  particular  difficulties  and  applying  the 
proper  stimulus  which  will  enable  him  to  "  find  himself." 

The  Assignment.  —  i.  Explanation  of  the  method  of  add- 
ing polynomials. 

2.   Recognition  of  the  new  problem  and  its  attendant  rule. 

Explanation  of  How  to  Add  Polynomials.  As  in  arithmetic 
we  can  only  add  or  subtract  terms  of  the  same  kind,  so  in 
algebra  like  must  come  under  like,  before  we  can  add  or  sub- 
tract. In  the  example,  Add:  a+^y  and  20,—  $y,  we  write  it 
as  follows  : 


30-  y 

and  add  each  term  separately.  Thus  a  plus  20  equals  30,  and 
4y  plus  minus  $y  equals  minus  y.  If  the  order  were  different 
in  the  example,  we  would  be  obliged  to  rearrange  the  terms 
so  that  the  a's  would  come  under  the  a's,  and  the  /s  under  the 

y's. 

Write  another  similar  example  on  the  board  and  ask  some- 
one to  direct  the  work.  Put  on  another  and  send  someone  to 
the  board  to  work  it.  If  all  claim  to  understand  the  opera- 
tions, pass  on  to  the  development  of  the  problem  involved  in 
this  lesson. 

Recognition  of  the  New  Problem  and  Its  Rule.  Ask  what 
kind  of  expressions  these  are  that  the  pupils  have  been  manipu- 
lating. After  you  get  the  right  term,  polynomials,  ask  what 
is  being  done  with  them.  Then  ask  someone  to  state  the 
problem  of  to-day's  lesson.  The  answer  should  be  "  Addition 
of  Polynomials." 

It  must  not  be  forgotten  that  every  day's  lesson  should 
have  an  object  or  problem;  to-day  it  is  Addition  of  Poly- 


Divisions  of  Elementary  Algebra  71 

nomials.  Various  schemes  may  be  used  to  emphasize  it. 
One  which  has  been  used  with  success  is  writing  it  upon  the 
board  with  yellow  crayon.  Thus  it  stands  emblazoned  in  the 
mind  of  the  child ;  and,  noticing  it  a  number  of  times  during 
the  hour,  he  cannot  forget  that  there  is  a  definite  object  in 
view,  and  that  the  work  in  hand  leads  up  to  its  understanding 
and  solution.  Again,  if  the  special  problem  under  consider- 
ation has  not  been  thoroughly  mastered,  it  remains  upon  the 
board  and  in  this  way  the  aim  of  the  lesson  is  even  more  deeply 
imprinted  upon  the  pupils'  understanding.  Pupils  like  to 
advance  and  if  they  find  that  another  day  must  be  spent  upon 
some  topic,  —  say  addition  of  monomials  —  because  they  did 
not  master  it,  renewed  efforts  will  be  made  to  move  on  to  some- 
thing new. 

Develop  the  rule  for  adding  polynomials  by  some  such 
analytical  method  as  this :  ask  why  we  put  the  various  terms 
involving  a  in  one  column,  and  what  kind  of  terms  these  are. 
Develop  the  definition  of  similar  terms.  When  the  example 
has  been  set  down,  what  do  we  do?  Draw  a  line  and  add  each 
column,  connecting  them  with  their  signs.  The  rule  has  thus 
been  developed.  Have  some  pupils  state  it  in  full.  Repeat 
with  different  ones  until  you  have  something  like  this : 

RULE.  — Arrange  the  similar  terms  in  the  same  column,  add 
each,  and  connect  the  resulting  terms  by  their  proper  signs. 

Work  two  or  three  examples  on  the  board,  asking  pupils 
to  apply  this  rule  by  specific  reference  to  the  terms  in  the  ex- 
ample thus  solved.  Such  mechanical  drill  is  necessary  in  all 
study  of  mathematics  but,  after  all,  there  is  only  one  object  of 
drill,  i.e.  to  grasp  the  principle  involved,  —  and  if  by  any  means 
this  may  be  done  quickly,  it  ought  in  all  justice  to  be  employed. 
Mathematics  should  not  become  a  master  but  a  servant. 


72      Supervised  Study  in  Mathematics  and  Science 

The  Study  of  the  Assignment. — I  or  Minimum  Assignment. 
Exercises  2-23,  in  textbook. 

II  or  Average  Assignment.    Exercises  24-27,  in  textbook. 

///  or  Maximum  Assignment.  Exercises  14-20,  on  page  41, 
Durell's  School  Algebra.1 

Verification.  —  Especial  attention  should  be  given  to  the 
pupils  who  were  yesterday  on  the  minimum  list.  For  ex- 
pediency, the  back  of  the  assignment  sheet  used  yesterday 
might  be  employed  to  record  the  names  of  these  pupils.  It 
might  be  a  good  plan,  as  soon  after  the  organization  of  the 
class  as  possible,  to  reseat  the  pupils,  placing  those  habitually  in 
the  minimum  classification  at  the  front  of  the  room  where 
they  may  be  easily  watched.  Care  should  be  taken,  however, 
not  to  name  the  groups  in  such  a  way  as  to  embarrass  any 
pupil.  The  teacher  will  use  tact  under  all  circumstances. 
Certainly  a  gain  in  facility  of  class  management  should  not 
be  achieved  at  the  loss  resulting  from  humiliating  or  embar- 
rassing any  member  of  the  class. 

If  the  teacher  is  unable  to  locate  a  pupil's  particular  diffi- 
culty on  account  of  illegible  figures  or  general  confusion  of  data, 
such  a  pupil  may  be  sent  to  a  side  blackboard  and  there  given 
a  private  lesson  in  some  of  the  fundamentals  of  study.  If  he 
seems  to  have  no  conception  of  the  problem,  let  him  analyze 
it  for  the  teacher,  telling  him  what  each  term  is  and  what  it 
signifies.  Make  him  comprehend  the  make-up  of  each  term  of 
the  expression ;  ask  him  why  he  puts  it  in  a  certain  place,  why 
he  draws  the  line  under  it,  how  he  treats  signs  in  adding,  etc. 

Such  individual  instruction  takes  time  but  is  well  worth 
while  if  thereby  some  boy  or  girl  is  saved  from  failure.  One 
or  two  such  private  lessons  like  this  each  day,  while  the  class 
1  Charles  E.  Merrill  Co. 


Divisions  of  Elementary  Algebra  73 

is  at  work  upon  the  assignment,  will  serve  to  keep  the  teacher 
pretty  busy,  and  yet  the  intensified  effort  will  well  repay  the 
expenditures  of  time  and  trouble.  The  satisfaction  of  joy 
over  seeing  the  weak  pupil  become  strong  is  a  great  reward  in 
itself.  Such  a  case  is  like  the  physician's.  It  needs  special 
study,  painstaking  oversight.  But  surely  the  restoration  to 
health  and  the  happy  development  of  a  "  case  "  is  a  deep 
professional  satisfaction. 

LESSON  VII 

UNIT  OF  INSTRUCTION   X.  —  THE  EQUATION  AND 
PROBLEMS 

LESSON  TYPE.  —  AN  EXPOSITORY  AND  How  TO 

STUDY  LESSON 
Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.    Simple  equations. 

Method.  Give  a  speed  test  in  reviewing  simple  equations. 
Have  a  large  assortment  of  exercises  on  the  board,  or  on 
mimeographed  sheets;  see  to  it  that  the  class  is  provided 
with  paper  and  pencils ;  set  all  at  work  on  the  minute.  See 
how  many  exercises  can  be  worked  correctly  in  some  definite 
length  of  time,  say  five  minutes.  When  the  time  is  up,  have 
all  stop  immediately.  Read  the  answers  and  ascertain  how 
many  attained  a  mark  of  100.  Collect  all  the  papers  and  put 
the  names  of  those  having  all  correct  upon  the  board.  In 
looking  over  the  papers  which  fall  under  100,  the  teacher  can 
note  just  where  the  trouble  lies  with  the  individual  pupils  and 
can  remedy  it  the  first  chance  he  has. 


74      Supervised  Study  in  Mathematics  and  Science 

Such  a  review,  based  on  the  time  element,  serves  to 
strengthen  the  pupil's  ability  to  concentrate,  puts  snap  into 
the  work,  and  lends  an  element  of  interest,  as  all  young  people 
like  anything  that  savors  of  a  contest. 

There  are  a  number  of  excellent  standardized  algebra  tests 
now  on  the  market,  which  the  teacher  may  use  to  advantage 
in  this  work.  These  tests  were  primarily  constructed  that 
there  might  be  given  an  opportunity  to  teachers  to  compare 
the  work  done  in  their  classes  with  that  in  other  school  systems. 
Since  it  is  a  fact  that  teachers  will  differ  more  or  less  markedly 
in  their  ordinary  grading  of  examination  papers  and  in  their 
judgment  of  pupils'  ability,  the  employment  of  tests,  which 
have  been  used  by  a  large  number  of  teachers  and  the  results 
of  which  have  been  standardized,  gives  an  excellent  means 
of  evaluating  the  work  in  any  class.  But  aside  from  this, 
these  tests  have  other  values  for  the  teacher.  They  point  out 
beyond  doubt  where  weaknesses  exist  and  allow  a  scientific 
basis  for  constructive  work.  Again,  they  arouse  a  vital  inter- 
est in  the  results  among  the  pupils  tested  because  they  like  to 
know  how  their  progress  compares  with  that  of  other  schools. 
The  avidity  with  which  pupils  will  strive  to  raise  the  standard 
of  the  school  and  of  themselves  offers  the  best  inducement 
for  intensified  work  in  the  classroom. 

(a)  The  Rugg  and  Clark  tests.  These  consist  of  booklets, 
containing  a  series  of  sixteen  tests  on  the  various  types  of 
algebraic  operations  from  one  on  collecting  terms  to  one  on 
quadratic  equations.  These  may  be  given  at  one  time  at  the 
completion  of  the  work  in  algebra,  but  the  author  has  found 
them  of  greater  value  in  checking  up  his  pupils  on  the  com- 
pletion of  each  type  process  and  noting  wherein  the  pupils  are 
weak,  thus  affording  an  opportunity  for  his  diagnosing  each 


Divisions  of  Elementary  Algebra  75 

individual  case  and  permitting  him  to  give  more  drill  in  those 
processes  that  seem  to  need  it. 

(6)  The  Hotz  scales.  These  are  in  the  form  of  sheets 
covering  all  the  processes  in  algebra,  including  problems.  The 
sheet  on  addition  and  subtraction  gives  examples  in  adding 
and  subtracting  terms,  expressions,  fractions,  and  radicals. 
The  one  on  equations  and  formulas  gives  examples  in  simple 
equations,  simultaneous  equations,  fractional,  radical,  and 
quadratic  equations,  and  equations  involving  the  manipula- 
tion of  formulas.  There  are  other  sheets  treating  multiplica- 
tion and  division,  problems,  and  graphs.  These  tests  are 
primarily  useful  in  testing  a  class  at  the  close  of  the  work  in 
algebra,  as  a  means  of  comparison  with  standard  scores.  Used 
in  connection  with  the  Rugg  and  Clark  tests,  they  form  a 
valuable  system  of  accurately  and  scientifically  testing  the 
progress  of  the  class. 

The  Assignment.  —  i.  Explanation  of  algebraic  repre- 
sentation. 

2.  The  algebraic  equations  applied  to  a  concrete  problem. 

3.  Definite  rules  for  studying  and  solving  problems. 

4.  Analyses  of  several  simple  problems. 

The  Representation  of  Concrete  Things  Algebraically.  Some 
such  questions  as  the  following  lead  up  very  logically  to  the 
study  of  the  problem  by  means  of  algebraic  representation : 

1.  Express  the  sum  of  five  and  three ;  of  a  and  b. 

2.  Express  the  difference  of  five  and  three ;  of  a  and  b. 

3.  What  number  increased  by  three  is  equal  to  eight  ? 

4.  What  number  diminished  by  three  is  equal  to  two  ? 

6.  How  do  the  last  two  questions  differ  from  the  first  two? 

The  pupils  will  see  that  in  the  last  two  questions  something 
is  lacking  which  is  to  be  found.  Tell  them  to  indicate  this 


76      Super-vised  Study  in  Mathematics  and  Science 

unknown  by  some  letter,  as  x.  Then  the  third  question  stated 
in  terms  of  the  known  values  and  the  unknown  values,  will 
read: 


and,  after  solving  by  transposition  of  terms, 

*=s; 

the  fourth  question  will  read  : 

3-3  =  2, 
or,  after  solving, 

*  =  5- 

6.  If  a  pencil  costs  five  cents,  what  will  three  cost? 

7.  If  a  pencil  costs  n  cents,  what  will  three  cost  ? 

8.  Express  the  fact  that  a  tablet  costs  five  cents  more  than  a 
pencil  in  both  6  and  7. 

9.  Express  the  fact  that  two  pencils  and  a  tablet  cost  fifteen 
cents.    Ans.  2^+5  =  15. 

10.   Solve  and  find  the  cost  of  one  pencil.    Ans.  «  =  5. 

These  questions  and  others  of  like  nature  may  be  read  to 
and  be  answered  by  the  class;  the  result  will  be  that  the 
pupils  will  gradually  sense  the  fact  that  by  using  literal  num- 
bers, we  are  able  to  represent  many  things  in  a  manner  that 
we  could  not  do  otherwise.  When  definite  values  are  assigned 
to  letters,  so  that  they  will  for  the  moment  stand  for  some- 
thing concrete,  they  take  on  an  entirely  different  meaning. 
Very  strange  "43;  plus  5^"  may  sound  to  a  boy,  but  when  we^ 
let  x  stands  for  dollars  it  assumes  a  very  sensible  and  familiar 
form.  Bring  out  the  fact  that  the  mechanical  work  preceding 
the  study  of  problems  has  aimed  at  enabling  the  class  to 
manipulate  the  resulting  algebraic  representation  of  some- 
thing concrete. 


Divisions  of  Elementary  Algebra  77 

Application  of  the  Equation  to  a  Concrete  Problem.  Let  us 
take  this  simple  equation,  x  plus  5  equals  12.  Have  someone 
analyze  it.  It  means  that  5  added  to  some  number  unknown 
will  give  us  12.  By  the  law  of  transposition  of  terms  in  equa- 
tions we  solve,  and  x  equals  7. 

Now  suppose  we  have  this  problem:  What  number  in- 
creased by  5  will  be  equal  to  1 2  ?  What  are  we  trying  to  find  ? 
A  certain  number.  Then  since  this  is  unknown,  we  will  for 
the  moment  let  x  stand  for  it  or  equal  it.  How  do  we  repre- 
sent increased  value?  By  adding.  Then  how  may  we 
indicate  the  expression  "number  increased  by  5"?  Since 
x  stands  for  the  number,  it  will  be  x+$.  But  according  to 
the  remainder  of  the  statement,  it  is  equal  to  12;  then 
#+5  =  12.  And  solving,  we  find  that  x  equals  7,  or  what 
we  wanted  to  find. 

In  like  analytical  manner  take  up  several  similar  problems, 
such  as :  What  number  diminished  by  or  increased  by  or 
exceeded  by,  etc.,  equals  something?  Lead  the  pupil  in  each 
case  to  see,  through  a  prior  arithmetical  representation  if 
necessary,  the  algebraic  representation  of  the  same. 

Definite  Directions  for  Solving  Problems.  At  this  point 
either  give  the  pupils  the  following  definite  steps,  previously 
mimeographed,  or  have  them  written  upon  the  board  and 
copied  by  the  pupils. 

a.  Read  the  problem  very  carefully;  study  it  until  you  know 
its  every  meaning.     Close  the  book  to  see  whether  you  can  state 
it  to  yourself,  silently.    Then  reopen  the  book  to  see  whether  you 
were  right. 

b.  Decide  what  is  the  thing  wanted  and  represent  it  by  x. 

c.  If  more  than  one  unknown  is  involved,  represent  them  by 
some  other  letters. 

d.  Express  in  algebraic  language  each  of  the  conditions  men- 
tioned in  the  problem. 


78      Supervised  Study  in  Mathematics  and  Science 

e.  Make  an  equation  of  the  two  statements  that  express  the 
same  conditions. 

/.   Solve  for  the  unknown. 

g.  There  should  be  as  many  equations  as  there  are  unknown 
quantities. 

Illustration  of  the  Directions.  Given  this  problem :  What 
number  diminished  by  8  is  equal  to  12? 

By  a :  We  mentally  analyze  this  to  be :  What  number  is  there 
which  will  be  equal  to  12  or  become  12  after  8  has  been  taken  away  ? 

By  b :  Number  is  the  thing  wanted ;  therefore  let  x  equal  the 
number. 

By  c :    Only  one  unknown  is  implied  in  this  problem. 

By  d:  x  minus  8,  and  12  are  two  expressions  concerning  the 
unknown. 

By  e :    They  are  equal,  therefore, 

x— 8=12. 
By  /:  x=  20,  or  what  was  desired. 

Analyses  of  Several  Problems.  The  teacher,  through  dif- 
ferent pupils,  will  then  work  out  in  similar  analytical  form,  the 
analyses  of  several  related  problems.  Insist  on  the  above 
mentioned  steps  being  followed  each  time;  pupils  must  be 
taught  how  to  study  problems  and  not  merely  how  to  solve  them. 
The  pith  of  the  whole  thing  lies  in  the  ability  of  the  pupil  to 
read  the  problem  intelligently  and  to  understand  it  so  thor- 
oughly that  he  can  tell  it  in  his  own  words.  Pupils  are  apt  to 
commence  work  before  they  fully  comprehend  what  is  given 
and  what  is  wanted. 

The  Study  of  the  Assignment.  —  /  or  Minimum  Assignment. 
Exercises  1-16,  in  text.  (All  of  these  should  have  been 
analyzed  in  class  but  not  worked.) 

II  or  Average  Assignment.  Exercises  17-21,  in  text.  (These 
have  not  been  analyzed.) 


Divisions  of  Elementary  Algebra  79 

III  or  Maximum  Assignment.  Exercises  26-30,  page  176, 
Vosburgh  and  Gentleman's  Junior  High  School  Mathematics, 
Second  Course. 

After  this  preliminary  lesson  on  how  to  study  problems, 
all  of  which  should  be  very  similar  and  not  too  difficult,  it  is 
advisable  to  take  up  problems  in  the  following  manner : 

Have  each  day  a  typical  problem  on  the  board ;  as  soon  as 
the  class  assembles,  let  all  read  it  over  carefully  and  study 
it  for  a  few  minutes.  Then  call  on  someone  to  state  what  is 
wanted,  someone  else  to  state  the  expressions,  someone  to 
make  the  equation,  and  someone  to  solve  it.  Call  on  a  number 
of  different  members  to  explain  various  phases  of  it,  making 
the  problem  an  object  of  class  study  and  analysis.  If  the 
problem  is  simple  in  principle,  it  is  often  found  profitable  to 
have  someone  make  up  a  similar  problem.  This  method  of 
having  the  pupil  make  his  own  problem  and  then  solve  it  will 
be  found  an  excellent  means  of  getting  him  interested  in  this 
kind  of  work.  Before  the  end  of  the  year  the  problems  which 
pupils  will  make  up  by  themselves  will  astonish  the  most 
experienced  teacher.  Data  may  be  supplied  from  current 
events,  such  as  elections,  ball  games,  business  statistics,  etc. 

The  author  cannot  recommend  too  highly  this  method  of 
spending  each  day  a  few  minutes  on  a  problem  and  then  pro- 
ceeding with  the  regular  work.  It  serves  to  keep  the  principles 
of  solving  problems  ever  before  the  class  rather  than  for  short, 
intermittent  periods.  Pupils  do  not  tire  of  them  but  will 
really  look  forward  to  this  phase  of  the  day's  work.  Solving 
problems  becomes  a  habit  and  what  is  quite  generally  con- 
sidered the  hardest  feature  of  algebra  loses  this  aspect,  be- 
cause the  children  have  become  so  used  to  problem  solving 
that  it  has  become ' '  second  nature. ' '  Occasionally,  an  advance 


8o      Supervised  Study  in  Mathematics  and  Science 

lesson  may  be  given  on  problems  only ;  but  the  above  plan 
has  been  found,  after  careful  trial,  to  cover  their  treatment 
adequately  and  well. 

LESSON  VIII 
UNIT  OF  INSTRUCTION  VH.  —  FACTORING 

LESSON  TYPE.  —  A  SOCIALIZED  LESSON 

Program  or  Time  Schedule 

The  Review 30  minutes 

The  Assignment 30  minutes 

The  Review.  —  Subject  Matter.    Factoring ;  all  cases. 

Method.  Have  the  boy  and  girl,  who  received  the  highest 
mark  on  the  last  grade  card  in  algebra,  choose  sides.  When 
this  has  been  done,  the  lines  should  be  placed  as  in  the  old 
fashioned  "  spelling-down  bee."  Commencing  with  the 
leaders  and  taking  them  alternately  from  the  two  sides,  the 
teacher  sends  the  pupils  in  turn  to  the  board  to  work  an 
exercise  in  factoring.  Excellent  material  will  be  found  in  the 
numerous  textbooks  on  the  teacher's  desk.  If  the  exercise  is 
worked  correctly,  the  pupil  returns  to  his  place  in  the  line  and 
one  from  the  opposite  side  goes  to  the  board.  If  a  pupil 
misses  an  exercise,  he  must  take  his  seat  and  during  the  re- 
mainder of  the  contest  work  out  all  the  exercises  on  paper  and 
hand  them  in  later  to  the  teacher. 

In  this  way  a  large  number  and  variety  of  exercises  may  be 
worked,  and  the  pupils  be  tested  for  their  skill  in  recognizing 
correct  answers,  for  it  is  evident  that  the  teacher  should  not 
take  too  active  a  part  in  reviews  of  this  kind.  The  pupils 
know  that  they  are  expected  to  pass  judgment  on  what  is 
worked  at  the  board.  If  serious  confusion  results,  the 
teacher  is  resorted  to  as  judge  of  the  court  of  appeals. 


Divisions  of  Elementary  Algebra  81 

When  one  side  wins,  i.e.  has  factored  down  its  opponent, 
the  assignment  is  taken  up. 

The  Assignment.  —  The  following  miscellaneous  exercises 
in  factoring,  taken  from  various  texts,  will  have  been  written 
upon  the  board.  The  first  eleven  will  illustrate  the  different 
type  forms  of  factoring,  and  a  question  or  remark  is  set 
opposite  each  to  direct  the  pupil  in  analyzing  it. 

Factor  : 

1.  3^~3X.  (Look  out  for  a  common  factor.) 

2.  a3  —  i.  (Difference  of  what?  Rule?) 

3.  0^  +  17^+72.  (What  type  form  have  you  here?) 

4.  ac  —  ax—  ibc+^bx.  (Be  careful  in  your  grouping.) 

6.  (x+a)2  —  (x—  a)2.  (The  parentheses  indicate  what  ?) 

6.  4a2+4fl6+62.  (Is  this  a  perfect  square?) 

7.  w8+w5.  (Note  the  powers  ;  odd  or  even?) 

8.  a4  +64  +a*62.  (How  may  this  be  made  a  perfect  square  ?) 

9.  rr3  —  7#+6.  (As  a  last  resort,  use  factor  theorem.) 

10.  r6—  s6.  (How  may   such    an   example   be  best 

treated  ?) 

11.  ac3"—  a35.  (Keep  in  mind  what  3«  and  36  are.) 

Then  give  the  pupils  an  equal  number  of  examples  illus- 
trating all  the  various  phases  of  factoring  but  in  a  different 
order,  of  course,  and  with  no  remarks.  Tell  the  pupils  to 
indicate,  in  addition  to  the  answer,  the  type  form  which  each 
illustrates,  as: 


=  (a+6)  (cP-ab+W).    Ans.    Sum  of  cubes. 
xn+l  +x  =  x  (xn  +  1  )  .     Ans.    Common  term. 

Supply  the  pupils  with  copies  of  Wheeler's  Algebra  and 
refer  them  to  page  70.  Tell  them  to  select  all  the  exercises 
that  illustrate  some  type  form  which  is  mentioned,  such  as 


82      Supervised  Study  in  Mathematics  and  Science 

the  difference  of  squares,  and  note  them  on  their   papers 
without  working  them.     For  example,  Nos.  3,  6,  10,  etc. 

Then  make  a  list  of  all  the  examples  which  illustrate  some 
other  case  in  factoring,  and  so  on,  covering  all  the  principal 
cases.  This  may  be  carried  out  to  any  degree  the  teacher 
wishes,  the  idea  being  to  acquaint  the  pupils  thoroughly  with 
the  different  type  forms,  to  practice  judgment  in  associating 
the  example  with  its  type  form,  and  therefore  in  selecting  the 
method  which  must  be  applied  for  its  solution. 

LESSON  IX 
UNIT  OF  INSTRUCTION  IX.  —  FRACTIONS 

LESSON  TYPE.  —  A  DEDUCTIVE  AND  How  TO 
STUDY  LESSON 

Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.    Multiplication  of  fractions. 

Method.  Write  a  number  of  exercises  on  the  board,  illus- 
trating the  previous  lesson  on  fractions.  Send  two  pupils  to 
the  board,  telling  the  others  to  work  the  examples  on  their 
papers.  Then  let  the  two  at  the  board  work  the  same  example 
simultaneously.  The  one  who  solves  it  first  may  take  his 
seat.  As  soon  as  the  second  pupil  has  solved  it,  the  teacher 
sends  a  third  pupil  to  the  board  to  work  the  next  exercise  with 
him.  In  this  way  the  second  pupil,  who  had  trouble,  gets 
additional  drill.  If  this  method  is  continued  the  poorer  ones 
will  remain  the  longest  and  therefore  get  the  most  practice. 
Meanwhile  all  the  others  are  busy.  The  pupils  are  expected 


Divisions  of  Elementary  Algebra  83 

to  solve  the  problems  as  rapidly  as  possible,  the  teacher  during 
the  meantime  helping  those  at  the  seats  who  are  experiencing 
difficulty.  To  vary  this  method,  it  is  sometimes  well  to  allow 
the  one  remaining  at  the  board  to  choose  his  next  opponent. 

The  Assignment.  —  i.   Definitions  of  complex  fractions. 

2.   Directions  for  their  solution. 

Complex  Fractions  Defined.  A  fraction  containing  one  or 
more  fractions  in  the  numerator  or  denominator,  is  called  a 
complex  fraction.  For  example : 

x 


b 

In  other  words,  it  means  that  the  quotient  obtained  by 
dividing  x  by  y  is  divided  by  the  quotient  obtained  by  dividing 
a  by  b.  It  may  be  set  down  like  this : 

x  _  a 

y  '  b 

and  it  then  becomes  similar  to  fractions  in  to-day's  lesson. 
But  sometimes  the  numerator  of  the  complex  fraction  may 
itself  be  a  mixed  number  or  a  series  of  fractions  or  another 
complex  fraction,  in  which  case  it  becomes  necessary  to  follow 
out  certain  definite  directions.  These  are : 

a.  Simplify  the  numerator. 

b.  Simplify  the  denominator. 

c.  Divide  the  first  result  or  quotient  by  the  second. 
Illustration : 


84      Supervised  Study  in  Mathematics  and  Science 
Eya: 

Eye:  ^  or 


The  Study  of  the  Assignment.  —  I  or  Minimum  Assign- 
ment. Exercises  2-11,  in  text. 

//  or  Average  Assignment.    Exercises  12-17,  in  text. 

///  or  Maximum  Assignment.  Make  up  and  solve  five 
complex  fractions. 

The  Silent  Study.  —  The  three  steps  as  outlined  above 
should  be  written  upon  the  board  with  colored  crayon  so  that 
the  pupils  may  have  them  plainly  in  view.  If  they  will  care- 
fully follow  out  each  step  as  applied  to  each  exercise,  the  class 
will  have  no  difficulty.  If  they  do  have  trouble,  it  will  be  from 
carelessness.  Most  of  the  difficulty  in  fractions  comes  from 
the  pupil's  own  illegible  figures.  The  exercises  on  account  of 
the  awkward  shape  of  their  graphic  representation  are  easily 
confused  unless  care  be  taken  to  set  them  down  in  good 
form  and  adhere  to  logical  order  in  their  solution.  The 
teacher,  by  passing  around  among  the  pupils,  will  be  able  to 
note  any  such  errors  and  he  should  avail  himself  of  the  op- 
portunity to  correct  them. 

In  case  someone  has  difficulty  and  calls  upon  the  teacher  for 
directions,  unless  the  difficulty  is  easily  found,  it  will  be  better 
for  him  to  start  anew,  with  the  teacher  overseeing  that  appli- 
cation is  made  of  the  three  successive  steps.  If  any  help  is 
given,  it  should  be  only  to  direct  properly  the  application  of 
these  rules  to  the  exercises  under  consideration. 


Divisions  of  Elementary  Algebra  85 

LESSON  X 

A  RED  LETTER  DAY  PROGRAM  IN  THE  NATURE  OF  A 
FIELD  DAY 

I.  Parade.  —  Each  pupil  in  the  class  is  to  be  assigned  some 
rule  covered  in  algebra  during  the  first  term,  which  he  is  to 
recite  upon  being  called  to  the  front  of  the  room.     For  instance, 
the  teacher,  or  judge  of  the  parade  as  he  might  be  designated, 
will  announce:    To  add  two  algebraic  numbers.    The  pupil 
who  has  been  assigned  this  rule  will  come  forward  and  answer : 
"  If  they  have  like  signs,  add  the  absolute  values  and  prefix  the 
common  sign ;  if  they  have  unlike  signs,  find  the  difference  of 
the  absolute  values  and  prefix  the  sign  of  the  numerically 
greater."  (Milne.) 

When  all  have  been  called  on,  the  judge  might  award  a 
prize,  of  no  intrinsic  value,  to  the  pupil  who  made  the  best 
appearance  and  recited  the  rule  in  the  most  distinct  voice. 

II.  Races.  —  (Select  three  qualified  pupils  to  act  as  judges.) 

1.  Multiplication  Race. 

Method.  Have  two  exercises  in  multiplication,  exactly  alike, 
upon  the  board.  Send  two  pupils  to  work  on  them ;  the  one 
getting  his  done  first  and  correctly  wins. 

2.  Division  Race. 

Method.     Similar  to  above,  but  using  different  pupils. 

3.  Championship  Race  in  a  Multiplication  and  Division 
Contest. 

Method.  Give  each  of  the  winners  in  the  first  two  races 
the  same  exercise,  which  will  be  a  combination  of  multipli- 
cation and  division.  The  one  solving  it  correctly  is  considered 
the  champion,  and  if  deemed  advisable  may  be  awarded  some 
prize,  such  as  a  colored  ribbon. 


86     Supervised  Study  in  Mathematics  and  Science 

4.  Grand  Relay  Race  in  Removing  Parentheses. 

Method.  Put  two  exercises  involving  the  removal  of  a 
number  of  signs  of  aggregation,  upon  the  board.  Pick  out  a 
relay  team  of  at  least  as  many  pupils  for  a  side  as  there  will  be 
complete  operations.  Start  one  from  each  side  at  the  same 
instant  and  as  soon  as  one  operation  is  complete,  let  the  next 
in  order  take  his  place.  The  side  that  first  gets  the  exercise 
done  correctly  wins.  It  is  suggested  that  two  pupils  be  al- 
lowed to  choose  sides  for  this,  members  going  to  the  board  in 
the  order  chosen. 

HI.   Game  of  Factoring  (modeled  after  baseball). 

Method.  Select  two  teams  of  nine  pupils  each,  preferably 
of  pupils  who  have  not  taken  part  yet  in  the  program,  except 
the  parade.  These  again  may  be  chosen  as  noted  above  for 
the  relay  race.  Also  select  an  umpire.  Each  team  will  be 
composed  of  a  pitcher,  catcher,  etc.,  as  in  a  ball  game.  These 
positions  will  probably  be  best  selected  or  assigned  by  the 
teacher,  who  will  also  explain  the  duties  of  the  players  and 
the  rules  of  the  game. 

The  pitcher  will  read  the  exercises  in  factoring  which  will 
have  been  handed  to  him  by  the  teacher. 

The  catcher  will  try  to  tell  the  type  form  of  the  exercise  before 
the  batter  can  do  so. 

The  batter  will  tell  the  type  form  of  the  exercise  as  soon  as 
he  can.  For  instance,  if  the  example  is:  at+iab+b2,  he  will 
say:  "  a  perfect  square." 

Each  baseman  and  fielder  will  have  been  assigned  some  type 
form  and  his  duty  will  be  to  solve  the  exercise  by  its  application 
as  soon  as  the  batter  refers  it  to  him.  For  instance,  the  short- 
stop may  have  been  assigned  "  a  perfect  square  "  as  his  position, 
so  that  as  soon  as  the  batter  gave  the  exercise  this  classification, 


Divisions  of  Elementary  Algebra  87 

the  shortstop  will  solve  it.  If  he  correctly  solves  it,  the 
batter  is  out ;  if  he  cannot  solve  it,  the  batter  makes  a  home 
run. 

If,  however,  the  catcher  gives  the  correct  form  before  the 
batter  does,  and  the  fielder  can  solve  it  correctly  the  batter  is 
also  out ;  if  the  fielder  fails  in  this  case,  it  is  called  a  strike  and 
the  batter  has  another  chance.  Three  such  strikes  will  put 
him  out. 

Again,  if  the  batter  gives  the  wrong  type  form,  it  counts 
a  strike.  Failure  to  understand  the  exercise  at  the  first  read- 
ing constitutes  a  foul ;  the  first  two  count  as  strikes,  as  in 
baseball. 

The  game  may  be  varied  as  to  number  of  innings,  accord- 
ing to  the  length  of  time  that  is  available,  but  probably  three 
will  suffice  for  this  program. 

As  already  mentioned,  there  should  be  an  umpire  to  call 
strikes,  fouls,  etc.  The  teacher  himself  may  act  as  referee 
in  case  of  dispute.  One  or  two  score  keepers  may  also  be 
selected. 

IV.  Picnic.  —  Method.  Each  of  the  following  typical 
exercises  in  fractions  may  be  considered  to  represent  different 
articles  of  food,  and  the  ability  to  solve  them  correctly  will 
give  the  pupil  a  helping  of  each  kind.  Inability  to  solve  the 
last  one,  for  instance,  which  represents  ice  cream,  would  de- 
prive him  of  this  dish.  All  pupils  are  given  paper  and  told  to 
solve  the  exercises  which  are  placed  upon  the  board.  In 
parentheses  is  indicated  what  each  exercise  represents.  The 
picnic  may  be  held  after  school,  if  the  period  is  not  long 
enough,  which  probably  will  be  the  case.  It  would  also  be 
difficult  to  determine  the  amount  of  food  needed  before 
then. 


88     Supervised  Study  in  Mathematics  and  Science 

EXERCISES 


Simplify :     -  +  i  -\ 

5  1+5 


Simplify  : 


Simplify  : 


Simplify  : 


Simplify  : 


X 


C  +  2        l6~Cz 


l  +  s 
ho 


Wo 


Reduce  to  mixed  number : 


P-6i+i$ 


*  —  I 


(Cabbage  salad) 
(Doughnuts) 

(Lemonade) 

(Cake) 
(Ice  cream) 


LESSON  XI 

UNIT  OF  INSTRUCTION   XIV.  —  RADICALS 

LESSON  TYPE.  —  A  SOCIALIZED  REVIEW 

Program  or  Time  Schedule 

The  Review 60  minutes 

The  Review.  —  Subject  Matter.  Reduction  of  radicals  to 
the  same  order. 

Method.  The  idea  of  an  occasional  socialized  lesson  is  to 
keep  up  the  interest,  throw  the  responsibility  of  failure  upon 
the  pupils,  and  impress  them  with  the  fact  that  algebra  may  be 
treated  from  the  standpoint  of  practical  application  to  the 
world's  work. 


Divisions  of  Elementary  Algebra  89 

The  class  is  divided  into  two  groups,  one  of  which  will  con- 
stitute a  sort  of  court  and  the  remainder  the  witnesses.  The 
teacher  will  select  a  judge,  a  lawyer,  a  jury  of  three  and  a  court 
crier.  They  may  be  chosen  on  the  basis  of  scholarship,  the 
best  pupil  being  judge,  etc. 

The  judge  will  take  the  teacher's  chair.  His  business  is  to 
determine  the  fairness  of  procedure  and  to  see  that  the  rule 
of  reducing  radicals  to  the  same  order  is  carefully  carried  out. 
The  lawyer  is  to  question  the  witness  on  the  exercise  under 
consideration.  The  jurors  are  to  decide  on  the  correctness  of 
the  solution  as  given  by  the  witnesses.  The  court  crier  is  to 
assign  the  exercises  to  the  various  members  of  the  class.  The 
remaining  pupils  in  the  class  constitute  the  witnesses  who 
are  to  testify  to  their  ability  to  solve  the  exercises  which  are 
under  review. 

Procedure.  The  teacher  swears  in  each  officer  by  demand- 
ing his  duty,  thus  giving  all  officials  the  opportunity  to  display 
their  absolute  grasp  of  the  principles  involved  in  the  work. 

Answers  to  the  following  or  similar  questions  might  be 
required  — 

Of  the  judge: 

"  State  the  rule  for  reduction  of  radicals  to  the  same  order. " 

"  Will  you  allow  the  lawyer  to  ask  questions  which  might  be 
misleading?" 

"  Will  your  attitude  toward  the  witness  be  austere  or  sympa- 
thetic?" 

"  Will  you  keep  order  in  the  court  room?" 

Of  the  lawyer: 

"What  is  your  duty?" 

"  Will  you  ask  helpful  questions  or  try  to  confuse  the  witness?" 

"  Will  you  act  on  the  principle  that  the  witness  is  innocent  of 


go      Supervised  Study  in  Mathematics  and  Science 

the  charge  that  he  cannot  solve  the  exercise  until  he  is  proven 
guilty?" 

Of  the  jurors: 

"  Will  you  promise  to  render  a  fair  decision  as  you  understand 
the  rules  of  algebra  relating  to  this  subject?" 

Of  the  court  crier: 

"  Will  you  state  the  exercise  in  a  clear  and  distinct  voice?" 
"  Will  you  be  impartial  in  your  assigning  of  exercises?" 

If  all  the  above  questions  are  answered  satisfactorily,  the 
officers  take  their  respective  stations  and  the  court  is  declared 
open  by  the  teacher  and  the  program  turned  over  to  the  judge, 
always  subject,  however,  to  recall  by  the  teacher. 

The  court  crier  calls  on  some  pupil,  who  goes  to  the  board 
and  is  given  this  exercise : 

Which  is  greater,  V$  or  -^6? 

The  witness  then  solves  the  exercise.  The  lawyer  may  ask 
him  any  question  on  his  work  or  the  method  of  solution,  the 
object  being  to  ascertain  whether  said  witness  thoroughly 
understands  the  exercise  and  is  able  to  acquit  himself  of  any 
inference  as  to  guilt  of  the  lack  of  such  knowledge.  The 
lawyer,  for  instance,  might  ask  him  why  he  cannot  compare 
them  as  they  stand,  or  what  is  the  significance  of  the  small 
figure  3. 

When  the  questions  and  the  solution  are  completed,  the 
jury  passes  judgment  on  the  pupil's  work  with  the  word  "  cor- 
rect "  or  "  incorrect "  as  the  case  may  be  If  the  verdict  is 
"  correct,"  another  witness  is  called  and  the  proceedings  re- 
peated. If,  however,  the  verdict  is  "  incorrect,"  the  judge 
must  pass  sentence.  He  may  order  a  new  trial  immediately 
or  he  may  impose  some  penalty,  such  as  isolation  in  some  part 


Divisions  of  Elementary  Algebra  91 

of  the  room  to  work  out  at  his  seat  some  extra  exercises  se- 
lected by  the  teacher.  When  all  the  class  have  been  examined 
in  turn,  the  failing  pupil,  or  pupils,  may  be  given  a  new  trial 
and  thus  offered  an  opportunity  of  redeeming  his  standing  in 
the  class.  If  he  again  proves  unable  to  clear  himself  of  the 
charge  of  "  guilty  of  error,"  the  judge  may  appoint  someone  to 
give  him  individual  aid.  The  court  may  remain  open  until 
all  the  class  have  been  cleared  of  any  imputation  of  lack  of 
ability  to  handle  these  exercises,  or  it  may  be  terminated  at 
any  time  by  the  teacher  and  court  declared  adjourned 
sine  die. 

Many  changes  will  occur  to  the  teacher  who  tries  out  this 
plan,  but  the  author  believes  that,  aside  from  the  added  inter- 
est given  to  the  solution  of  these  exercises,  the  pupils  are  mean- 
while unconsciously  learning  invaluable  lessons  in  court  pro- 
cedure and  social  responsibility  as  well  as  lessons  in  self- 
expression  and  self-control. 

It  is  sometimes  advisable  to  devote  the  entire  period  to  this 
kind  of  work.  Hence  the  assignment  of  exercises  for  further 
study  will  be  given  in  one  assortment  only  and  it  will  be  done 
outside  of  class  and  handed  in  the  next  day.  In  this  case, 
twenty-five  exercises  to  review  further  this  type  of  problems 
may  be  assigned,  taken  from  some  supplementary  textbook. 
In  order  to  carry  out  the  idea  of  the  three  assignments,  the 
first  ten  might  be  considered  a  minimum  number  to  be  solved, 
the  first  fifteen  the  average  number  and  the  entire  twenty- 
five  the  maximum  quota. 

The  Study  of  the  Assignment.  —  Twenty-five  similar 
exercises  taken  from  Ford  and  Ammerman's  First  Course  in 
Algebra,1  page  256. 

1  The  Macmillan  Company. 


92      Supervised  Study  in  Mathematics  and  Science 
LESSON  XII 

UNIT   OF   INSTRUCTION   XV.  —  QUADRATIC  EQUATIONS 

LESSON  TYPE.  —  AN  EXPOSITORY  AND  How  TO 
STUDY  LESSON 

Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.  Quadratic  equations  solved 
by  factoring. 

Method.  Have  an  example  of  this  type  written  on  each 
panel  of  the  blackboard  and  send  the  minimum  workers,  as 
developed  from  the  previous  day's  work,  to  solve  them.  Also 
place  over  them  as  monitors  the  maximum  pupils,  admonish- 
ing them  not  to  solve  the  exercises,  but  to  see  that  they  are 
worked  correctly.  The  teacher  may  meanwhile  ask  the 
other  pupils  necessary  questions  concerning  such  exercises. 
As  soon  as  the  exercises  on  the  board  have  been  completed  and 
pronounced  correct  by  the  monitors,  take  up  the  new  work, 
the  work  on  the  board  being  left  for  future  inspection. 

The  Assignment.  —  i.  Explanation  that  all  quadratic  equa- 
tions cannot  be  easily  factored  and  that  even  those  which 
can  be  factored  may  also  be  solved  by  other  methods. 

2.  Exposition  of  the  method  of  completing  the  square. 

3.  Statement  and  explanation  of  the  rule  for  completing  the 
square. 

Other  Methods  Sometimes  Necessary.  Write  a  quadratic 
equation  upon  the  board  which  cannot  be  factored  as  it  stands. 
For  example : 

yp  —  ix— 6=0. 


Divisions  of  Elementary  Algebra  93 

Method  of  Completing  the  Square.    Take  this  example  : 
x2—  x  —  2=0. 

It  is  in  a  form  that  can  be  factored  at  once  into  (#  —  2) 
(#+i)  ;  but  let  us  solve  it  as  though  it  could  not  be  so  factored. 

a.  First,  transpose  terms  so  that  the  unknowns  will  be  on 
one  side  and  the  knowns  on  the  other  side  of  the  equation. 

xz—  x  =  2. 

b.  Then,  unless,  as  in  this  case,  the  coefficient  of  the  un- 
known to  the  second  degree  is  unity,  divide  the  equation 
through  by  the  coefficient  so  that  it  will  be  unity. 

c.  Now  take  one  half  the  coefficient  of  the  unknown  to  the 
first  degree,  in  this  case  i,  and  square  it,  i.e.  |  squared  be- 
comes \. 

d.  Add  this  to  both  sides  of  the  equation,  as 


e.  Extract  the  square  root  of  both  members,  and  we  have 
*-|=*f 

/.   Solve  for  x;  x  =  2  or  —  i.     Ans. 

g.  Verify  by  substitution,  4  —  2  —  2  =  o,  or  4  =4,  and  i  +  1  — 
2  =o,  or  2  =  2. 

Statement  of  Rule  for  Completing  the  Square.  This  is  al- 
ready given  above,  but  divided  into  separate  steps  ;  it  may 
now  be  given  entirely. 

Take  a  similar  exercise  and  have  someone  tell  what  should 
be  done,  following  each  step  by  the  directions  above.  Repeat 
until  the  class  understands  this  new  lesson. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assign- 
ment. Exercises  3-20. 

//  or  Average  Assignment.    Exercises  21-25. 


94     Supervised  Study  in  Mathematics  and  Science 

III  or  Maximum  Assignment.  Exercises  18,  19,  21,  on 
page  269  of  Ford  and  Ammerman's  First  Course  in  Algebra.1 

The  Silent  Study.  —  Tell  the  pupils  to  apply  each  step  as 
in  the  above  set  of  rules,  and  to  letter  the  result  of  each  step 
with  its  corresponding  letter.  Their  solutions  should  appear 
as  follows  : 

EXERCISE:  2*+3*-27-o.  (Transposition) 

a.  2X2  +3^  =  27. 

b.  x*+—=—  •  (Division  by  coefficient) 

2  2 


.9=£+.S__+..    (Addition) 
2      16      2      16      16     16 


i6      16 

e.  *+f==fcJr--          (Extracting  square  root) 

/.  x=±£-,  or  —  ¥•  (Solving  for  x) 


*'  2(3)^l+l==ll'.  (Verification) 

If  the  pupil  has  trouble  and  it  proves  to  be  a  matter  of 
committing  small  errors  of  computation,  let  him  go  over  his 
work  again  and  carefully  check  each  step. 

If,  on  the  other  hand,  the  pupil  seems  to  have  failed  to 
grasp  the  principles  involved  and  to  be  unable  to  apply  the 
steps  as  suggested,  let  him  go  to  the  board  and  work  there 
under  the  teacher's  supervision.  Let  him  read  the  rule  over 
carefully  and  tell  just  what  each  step  means.  Let  him  study 
1  The  Macmillan  Company. 


Divisions  of  Elementary  Algebra  95 

it  until  he  does  understand  thoroughly  just  what  it  implies. 
A  little  direction  from  the  teacher  should  relieve  any  mis- 
understanding of  the  method. 

If  the  pupil  still  fails  to  grasp  the  procedure,  the  teacher 
should  solve  the  problem,  explaining  in  detail  each  step  as  he 
proceeds.  Then  erase  and  have  the  pupil  go  over  the  same 
process,  step  by  step.  Many  pupils  fail  to  grasp  thoroughly 
directions  when  given  to  a  class  at  large,  either  because  of 
inattention  or  failure  to  comprehend  each  step  as  it  is  de- 
veloped. This  individual  supervision  will  usually  bring  the 
desired  results.  When  the  pupil  finds  that  you  are  not  going 
to  work  his  problem  for  him  but  will  direct  him  how  to  do  it,  he 
will  increase  his  efforts  and  try  to  master  the  technic. 

LESSON  XIII 
TESTS 

LESSON  TYPE.  —  AN  EXAMINATION 

Program  or  Time  Schedule 
Examination 60  minutes 

Real  Tests.  —  The  daily  review  is  in  fact  a  continuous  test 
of  the  ability  of  the  pupil  to  do  his  work.  As  has  already  been 
emphasized,  the  pupil's  outside  work  is  important  only  in  so 
far  as  it  prepares  him  to  do  something  similar  again.  After 
a  pupil  has  worked  thirty  problems  in  addition  of  fractions, 
and  handed  them  in  to  the  teacher,  and  reported  no  difficulties, 
the  teacher  is  certainly  within  his  rights  when  he  expects  that 
pupil  to  be  able  to  do  similar  examples  under  his  observation 
in  class.  If  he  cannot,  then  his  outside  work  is  valueless  and 
can  be  safely  assumed  not  to  be  his  own.  The  pupil  should 
be  made  to  realize  that  his  assigned  work  is  to  make  him  pro- 


g6      Supervised  Study  in  Mathematics  and  Science 

ficient  in  a  certain  line  and  he  must  feel  that  the  real  test  of 
his  proficiency  is  not  measured  through  his  practice  but 
through  his  ability  to  repeat  the  process.  The  review,  there- 
fore, while  being  a  part  of  the  recitation  participated  in  by  all, 
should  be  especially  directed  toward  the  weaker  pupils. 

Then  the  ability  to  understand  the  new  work  from  day  to  day 
and  to  follow  directions  and  get  results  is  another  real  test  of 
the  pupil's  advancement.  It  takes  no  written  examination  to 
demonstrate  that  a  pupil  is  failing,  when  he  continually  comes 
to  class  unprepared  and  is  unable  to  comprehend  the  new 
work  as  it  is  developed.  In  fact,  the  rapidity  with  which  some 
grasp  the  new  topic  marks  them  specifically  as  unusual  pupils 
in  that  line. 

Problems  are  in  many  ways  real  tests  of  the  pupil's  ability. 
The  very  statement  of  the  problem  in  algebraic  terms  is 
indicative  of  logical  reasoning;  after  this  process  becomes 
familiar,  then  the  ability  to  solve  problems  so  stated  is  further 
indicative  of  the  pupil's  mentality  and  mental  growth.  These 
processes  characterize  the  whole  field  of  this  subject  and 
readily  and  accurately  measure  the  pupil's  ability.  Some 
pupils  claim  they  can  never  master  applied  problems,  and  some 
teachers  display  a  sympathetic  attitude  to  this  position  by 
stating  that,  except  for  problems,  their  pupils  could  pass  the 
final  examination.  While  it  is  a  fact  that  this  is  true  in  many 
cases,  yet  this  is  no  reason  why  problems  should  be  slighted ; 
in  fact,  it  seems  to  the  author  almost  an  unanswerable  argu- 
ment that  problems  should  comprise  the  major  part  of  the 
final  examination.  But  there  are  problems  and  problems ; 
all  preposterous,  ultra  complex,  and  catch  problems  should, 
of  course,  be  omitted.  They  should  be  fairly  simple,  fully 
reasonable,  and  straightforward  in  their  applications.  Some 


Divisions  of  Elementary  Algebra  97 

texts,  such  as  Ford  and  Ammerman,1  have  exceedingly  well- 
selected,  practical  problems  in  accord  with  the  best  modern 
conceptions  of  instruction  in  algebra.  The  ability  to  handle 
such  problems  constitutes  an  almost  ideal  test  of  the  pupils' 
knowledge  of  algebra. 

Besides  these  three  daily  records  which  themselves  constitute 
real  tests  of  the  ability  of  the  pupil  to  grasp  the  subject,  there 
remains  the  formal  written  examination  and  its  treatment. 

Written  Tests.  —  i.  Object.  The  object  of  the  written 
examination  is  twofold :  (a)  to  find  the  points  the  pupils  do 
not  understand,  in  order  to  drill  on  them,  and  (b)  to  obtain 
a  definite  grade  or  per  cent.  The  former  is  the  really  valuable 
thing ;  the  latter  is  immaterial. 

2.  Testing  for  Weak  Points.  The  teacher  wishes  to  know 
definitely  what  points  the  class  has  mastered  and  what  points 
have  mastered  the  class.  He  is  interested  in  the  former  only 
from  an  academic  standpoint,  but  in  the  latter  he  is  vitally 
interested  because  it  is  his  business  not  only  to  diagnose  but  to 
treat  and  cure. 

Hence,  after  each  unit  of  instruction  has  been  completed, 
a  review  test  is  given  to  find  out  what  the  class  has  failed  to 
grasp.  These  questions  should  be  in  the  nature  of  drill  ex- 
ercises, similar  in  nature  to  those  already  studied  and  given 
solely  for  the  reason  stated  above.  The  writer  has  found  it 
expedient  and  trustworthy  to  use  some  standardized  test  in 
which  the  results  may  be  compared  with  a  standard  of  excel- 
lency, rather  than  to  use  exercises  of  his  own  making.  Rugg's 
tests2  are  exceedingly  good  for  this  work.  There  are  others 
on  the  market,  mentioned  in  Lesson  VII.  When  the  class 

1  "First  and  Second  Course  in  Algebra" ;  The  Macmillan  Company,  1919. 

2  University  of  Chicago  Press. 


98     Supervised  Study  in  Mathematics  and  Science 

fails  to  come  up  to  the  standard  set  in  these  tests,  the 
pupils  who  fall  below  should  be  noted  and  special  attention 
given  to  them  on  the  principles  in  which  they  are  found 
to  be  weak.  Meanwhile  those  who  have  measured  up  to 
standard  should  be  given  more  advanced  work  along  the  same 
line. 

The  principle  of  the  minimum-average-maximum  classifi- 
cation should  ever  be  kept  in  mind,  and  the  teacher  should 
strive  not  only  to  get  all  to  qualify  on  the  minimum  require- 
ments but  also  should  assist  those  who  are  capable  to  master 
more  difficult  applications  and  get  higher  grades.  Teachers 
as  well  as  pupils  are  very  apt  to  be  satisfied  with  the 
former. 

3.  Testing  for  a  Final  Grade.  The  ideal  method  of  giving 
grades  is  the  average  class  mark.  The  written  final  examina- 
tion, however,  is  bound  to  be  with  us  for  some  time  to  come, 
in  at  least  some  modified  form.  It  should  be  along  the  line  of 
questions  which  test  reasoning  ability  and  not  the  memory 
alone.  As  already  noted,  problems  offer  the  most  ideal  method 
of  formally  testing  the  ability  to  "do  algebra,"  but  questions 
of  other  kinds  may  be  asked  which  will  approximate,  at 
least,  the  standard  sought  in  the  above  statement.  A  sample 
examination  of  such  nature  follows,  by  way  of  illustration  or 
suggestion,  covering  the  work  of  the  first  term. 

A  SUGGESTED  EXAMINATION 

DIRECTIONS  :  Answer  all  the  questions  under  Part  I.  Necessary 
to  pass,  3  correct  from  Part  I  and  2  correct  from  Part  II.  If  all 
of  Part  I  and  Part  II  are  correct,  the  grade  will  be  85%.  If,  in 
addition,  2  from  Part  III  are  correct,  the  pupil  will  receive  honors. 


Divisions  of  Elementary  Algebra  99 

PART  I 

{Answer  all  the  questions} 

1.  Explain  what  you  understand   by   positive   and   negative 
numbers,  using  some  illustration,  such  as  the  thermometer,  a  bank 
account,  etc.     Give  the  rules  and  explanations  of  how  to  add,  sub- 
tract, multiply,  and  divide  signed  terms. 

2.  Make  up  and  solve  an  example  involving  the  removal  of 
parentheses,  containing  at  least  twelve  terms  and  at  least  three 
groups  under  signs  of  aggregation. 

3.  Divide  x5+y5  by  x+y  giving  reasons  for  each  step,  and 
prove  your  answer. 

4.  Solve  the  following  exercise  in  cancellation,  stating   what 
typical  case  in  factoring  is  exemplified  in  each  step,  and  explaining 
the  process  involved  in  each  step: 

a4  -ft4      ^a?+b* 
a?-2ab+b2  '  o?-ab' 

6.  Replace  the  question  marks  with  values  for  x  and  y,  and 
solve  the  finished  example  : 


PART  II 

(Answer  at  least  two) 

6.  Simplify:  a3  -[a2  -(3  -a)  -\2+a*-(i-a)+a?\]. 

7.  Factor:   (a)    i-w8;   (b)    lox^-'jxy+y2',   (c)   a 

a2 
ac  -- 

8.  Simplify  :  -  —  i. 


--  ac 
4 


ioo    Supervised  Study  in  Mathematics  and  Science 

PART  III 
9.   Expand:   (.3^  — .4*  — 1.4)  (.2Z2+. 2#  — 2.5). 

10.  Divide:  a2*+2-az+164-2&2*+3&V!-1-c2*-2bya*+1+&z-ca!-1. 

11.  Factor:   a3  —  70 +6. 

12.  Simplify:     


a  — 


i 
a 


a—x 


It  will  be  noted  that  each  part  successively  comprises 
exercises  of  an  increasing  difficulty,  more  technical  and  assum- 
ing advanced  knowledge  and  ability  on  the  part  of  the  pupil. 

Value  of  This  Form  of  Examinations.  —  This  type  of  exam- 
inations seems  to  be  a  fairer  test  of  the  pupils'  ability  than 
the  conventional  one  which  comprises  a  series  of  exercises, 
any  combination  of  which  may  be  selected  and  the  grade  found 
by  marking  the  paper.  In  such  an  examination  there  is  really 
no  testing  of  the  ability  of  the  brighter  pupils  to  do  superior 
work  since  all  the  questions  are  aimed  at  the  average  student. 
On  the  other  hand,  there  is  no  minimum  requirement  for  the 
pupils  of  lesser  ability,  which  should  be  demanded  of  all.  In 
other  words,  the  pupils  tested  may  attempt  all  and  possibly 
salvage  enough  to  get  a  passing  grade,  while  their  mastery  of 
any  one  phase  may  be  below  standard. 

In  the  type  above  suggested,  it  is  hoped  that  these  malad- 
justments may  be  at  least  partially  eliminated.  In  the  first 
place,  a  certain  number  of  exercises  must  be  done  correctly  in 
order  to  receive  a  passing  grade.  These  may  be  selected  from 
Parts  I  and  II,  which  give  a  choice  in  each  part.  If,  therefore, 
the  pupil  cannot  solve  five  eighths  of  these  without  error,  he 


Divisions  of  Elementary  Algebra  101 

will  fail.  If,  in  addition  to  these  five,  he  succeeds  in  getting 
all  of  the  first  part  and  all  of  the  second  part  correct,  he  will 
get  a  higher  grade,  or  85.  And  if  there  be  anyone  who  can 
further  work  correctly  two  from  the  third  part,  he  will  receive 
honors.  Thus  each  type  of  pupils,  whether  of  minimum, 
average,  or  maximum  ability,  is  given  an  opportunity  of  re- 
ceiving higher  credit  for  additional  work  and  work  of  a  more 
advanced  kind.  But  none  can  get  honors  by  solving  more 
of  those  of  the  simpler  type  or  by  carefully  selecting  all  the 
easy  exercises  on  the  examination  paper.  So  it  would  seem 
that  this  type  of  examinations  definitely  tests  everyone  in  the 
class. 


SECOND   SECTION 
PLANE    GEOMETRY 


CHAPTER  THREE 

DIVISIONS   OF  PLANE   GEOMETRY 

Practically  all  texts  in  Plane  Geometry  agree  on  the 
Euclidian  arrangement  of  material  under  the  FIVE-BOOK 
system.  These,  with  the  introduction,  thus  become  the  units 
of  instruction. 

A.  UNITS  OF  INSTRUCTION. 

I.  Introduction. 

II.  Book  I.     Rectilinear  figures. 

III.  Book  II.    The  circle. 

IV.  Book  III.    Proportion.     Similar  polygons. 
V.  Book  IV.    Areas  of  polygons. 

VI.  Book  V.    Regular  polygons  and  circles. 

B.  THE  DIVISION  OF  EACH  UNIT  INTO  UNITS  OF  RECITATION. 

I.  INTRODUCTION.    The  material  in  this  unit  will  vary  in 

different  books  but  will  probably  comprise : 
Units  of  Recitation : 

1.  Definitions. 

2.  Axioms,  postulates,  etc. 

3.  Oral  exercises. 

4.  Historical  notes. 

II.  Book  I.    RECTILINEAR  FIGURES. 
Units  of  Recitation  : 

1.  Triangles. 

2.  Parallel  lines. 

3.  Loci. 

4.  Quadrilaterals. 

5.  Polygons. 

6.  Exercises  and  problems. 

105 


io6    Supervised  Study  in  Mathematics  and  Science 

HI.  Book  II.    CIRCLES. 

Units  of  Recitation  : 

1.  Theorems  on  the  circle. 

2.  Problems  on  the  circle. 

3.  Exercises. 

IV.  Book  HI.    PROPORTION.     SIMILAR  POLYGONS. 

Units  of  Recitation  : 

1.  Theorems  on  proportion. 

2.  Similar  polygons. 

3.  Exercises  and  problems. 

V.  Book  IV.    AREAS  or  POLYGONS. 
Units  of  Recitation  : 

1.  Areas  of  equivalent  and  similar  figures. 

2.  Exercises  and  problems. 

VI.  Book  V.    REGULAR  POLYGONS  AND  MEASUREMENT  OF 
CIRCLES. 

Units  of  Recitation: 

1.  Regular  polygons. 

2.  Measurement  of  the  circle. 

3.  Maxima  and  minima. 

4.  Symmetry. 

5.  Exercises  and  problems. 

LESSON  I 
THE  INSPIRATIONAL  PREVIEW 

Meaning  of  Inspirational  Preview.  —  As  has  already  been 
emphasized  in  the  companion  introduction  to  algebra,  the 
purpose  of  the  preview  is  to  arouse  the  child's  desire  to  learn 
geometry.  To  this  end  it  is  important  to  skillfully  advertise 
the  subject.  Such  advertising  should  include  flashes  from  the 
history  of  geometry,  well  proved  values  of  its  application  and 


Divisions  of  Plane  Geometry  107 

a  brief  survey  of  the  contents  of  the  course ;  just  enough  of 
each  to  whet  the  appetite  for  more.  The  pupils  are  likely  to 
feel  that  it  is  going  to  be  dry  or  hard  or  futile,  but  such  mis- 
conceptions may  be  quickly  removed  by  a  clear  preview. 

History  of  Geometry.  The  word  geometry  —  meaning  in  the 
Greek  language,  to  measure  land  or  earth  —  indicates  that  the 
science  developed  from  the  early  practice  of  the  modern 
science  of  surveying.  It  is  not  known  with  what  people  the 
science  originated  but  certainly  the  Egyptians  had  acquired  a 
considerable  understanding  of  the  subject  as  is  attested  by 
their  pyramids,  which  are  built  in  strictly  geometric  designs. 
Recently  discovered  tablets  have  proved  that  also  the  Babylo- 
nians were  acquainted  with  this  subject. 

But  the  first  practical  study  of  geometry  for  its  own  sake 
was  made  by  the  Greeks.  Pythagoras,  about  560  B.C., 
discovered  many  new  propositions  and  added  to  the  popularity 
and  inspired  increased  study  of  the  subject.  Euclid  was  the 
first  to  make  a  successful  attempt  to  write  a  book  which  would 
contain  in  an  orderly  manner  all  the  known  proofs,  and  so 
well  did  he  do  his  work  that  all  subsequent  texts  have  been 
modeled  after  his  book.  He  lived  between  330  and  275  B.C. 

So  we  find  that  geometry,  like  algebra,  is  the  combined 
product  of  many  minds  of  many  ages.  Contributions  are 
being  made  to  its  content  at  the  present  time. 

Practical  Value.  Geometry  has  a  vital  connection  with 
many  important  phases  of  life.  Indeed  it  is  difficult  to  com- 
prehend how  our  modern  civilization,  with  its  machinery, 
buildings,  bridges,  ships,  and  other  marvelous  engineering 
accomplishments,  could  exist  without  the  contributions  which 
this  science  has  made.  Without  a  knowledge  of  geometry  we 
would  know  nothing  about  the  size  of  the  earth,  about  our 


io8    Supervised  Study  in  Mathematics  and  Science 

solar  system,  about  the  universe  as  we  to-day  conceive  it. 
The  principles  of  geometry  are  used  by  engineers  in  construct- 
ing bridges,  trestle  work  of  all  kinds,  arches,  etc.  The  em- 
ployment of  formulas  developed  through  use  of  geometry  is 
universal  in  the  application  of  mathematical  knowledge  to 
all  kinds  of  mechanical  construction.  Such  structures  as  the 
Brooklyn  Bridge,  the  Eiffel  Tower,  the  Capitol  at  Washington, 
the  Ferris  Wheel,  the  Roosevelt  Dam,  and  countless  others, 
are  all  the  result  of  the  application  of  geometric  principles  to 
practical  engineering  accomplishments. 

Again,  geometry  is  made  use  of  in  designing  mosaics,  vault- 
ings, tile  patterns,  church  windows,  parquet  flooring,  steel 
ceilings,  oilcloth,  iron  grilles,  embroidery,  lace  work,  etc. 

Although  a  study  of  mensuration  begins  in  arithmetic,  it 
is  nevertheless  only  fair  to  say  that  its  derivation  is  purely 
geometrical,  and  this  science  should  be  credited  for  its 
great  contribution  to  this  practical  aspect  of  mathematics. 

If  we  believe  in  formal  discipline,  then  geometry  certainly 
deserves  much  credit  for  its  contributions  along  this  line. 
Plato  believed  explicitly  in  the  mental  value  of  this  subject 
and  it  is  said  that  he  had  a  sign  over  his  school  of  philosophy, 
reading,  "  He  who  knows  not  geometry  may  not  enter  here." 
Abraham  Lincoln  studied  geometry  to  cultivate  a  logical 
mind.  Geometry  is  practically  the  only  subject  in  the  school 
program  which  gives  practice  in  the  use  of  pure  deductive  logic. 
The  concentration  of  mind  and  the  method  of  logical  steps  re- 
quired to  prove  original  problems  in  geometry  combine  to 
give  one  of  the  best  mental  exercises  offered  by  any  subject 
in  school.  The  pupil  learns  to  think  clearly,  logically,  concisely, 
along  mathematical  lines.  Its  study  offers  more  possibilities 
for  correct  and  incisive  thinking  than  any  other  branch  of 


Divisions  of  Plane  Geometry  109 

mathematics.  If  more  of  the  method  of  geometrical  proof 
were  applied  in  situations  demanding  clear-cut  thinking,  it  is 
not  unlikely  that  many  of  the  present-day  issues  would  be 
better  understood. 

A  Bird's-eye  View  of  the  Course.  Call  the  attention  of  the 
pupils  to  the  fact  that  they  have  already  had  some  geometry 
in  arithmetic  since  they  have  studied  about  areas  of  plane  and 
solid  figures,  such  as  rectangles,  triangles,  circles,  cylinders, 
pyramids,  spheres,  and  cones.  In  algebra,  also,  some  of  the 
facts  of  geometry  have  been  employed  in  data  stated  in 
the  problems.  But  now  we  come  to  a  study  of  these 
truths  from  a  new  point  of  view,  that  of  their  value  in 
themselves  and  not  so  much  their  application  to  the  affairs  of 
the  world. 

Note  with  the  class  the  division  of  the  text  into  so-called 
books  or  chapters,  each  book  devoted  to  some  particular  phase 
of  the  subject.  Explain  that  these  are  the  books  of  Euclid, 
and  while  they  are  now  collected  in  one  textbook,  they  still 
retain  the  old  classification.  Call  the  pupils'  attention  to 
the  numerous  drawings  and  devote  a  few  words  to  an  explana- 
tion of  the  value  of  neat,  accurate  figures. 

In  conclusion  and  for  a  review  of  the  above  introductory 
work,  put  the  following  questions  upon  the  board  and  require 
the  answers  to  be  written  out  and  handed  in  the  following 
day.  Those  given  below  are  only  suggestive ;  better  ones  will 
no  doubt  occur  to  the  teacher. 

1.  The  first  syllable  of  geometry,  or  ge,  means  earth.     Can 
you  think  of  any  other  science  which  begins  with  get     If  so,  of 
what  is  it  the  study? 

2.  What   natural   phenomenon  occurring   regularly  in  Egypt 
caused  the  early  development  of  surveying? 


no    Supervised  Study  in  Mathematics  and  Science 

3.  Find  out,  by  reference  to  other  texts  in  geometry,  some 
other  men  than  Pythagoras  and  Euclid  who  contributed  to  the 
development  of  this  subject. 

4.  State  the  Pythagorean  theorem  which  you  studied  in  arith- 
metic. 

6.   How  does  geometry  help  us  to  a  knowledge  of  the  size  of  the 
earth  ?    Of  what  value  is  this  knowledge  to  us  ? 

6.  Mention  some  decorative  design  you  have  noticed  which 
is  composed  of  geometric  figures. 

7.  What  do  you  understand  by  deductive  reasoning? 

8.  You  have  learned  in  your  arithmetic  that  the  area  of  a 
rectangle  is  the  product  of  the  length  and  the  width.     How  did  the 
Egyptians  make  use  of  this  rule?    Has  it  any  value  in  modern 
outdoor  sports? 

9.  Into  how  many  books  is  the  work  in  geometry  divided? 
10.   Why  must  we  have  neatly  drawn  figures?    Are  they  more 

necessary  in  geometry  than  in  arithmetic  ?    Why  ? 

LESSON  II 

UNIT   OF  INSTRUCTION   II.  —  Book  I 
RECTILINEAR  FIGURES 

LESSON  TYPE.  —  A  DEDUCTIVE  AND  How  TO  STUDY 

LESSON 

Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.    Properties  of  angles. 
Method.     Draw  a  straight  line  upon  the  blackboard,  as 


Divisions  of  Plane  Geometry 


in 


Ask  what  a  straight  line  is. 
A 


Designate  a  point  by  C,  as 
B 


Ask  what  an  angle  is.  Have  someone  go  to  the  board  and 
draw  a  number  of  angles,  lettering  each  by  ACB  in  which  C 
is  the  vertex.  Ask  what  kind  of  angle  the  above  ACB  is. 
What  is  the  value  in  degrees  of  such  an  angle?  Draw  a  line 
to  C  from  some  other  point  outside,  as  P. 


What  kind  of  angles  are  ACP  and  BCP?  What  is  the 
value  of  the  two? 

Extend  PC  through  C  to  K.  What  kind  of  angles  are 
PC  A  and  KCA  ?  KCA  and  KCB  ? 

What  kind  of  line  is  PK?  What  kind  of  angle  is  PCK? 
Then  what  is  the  value  in  degrees  of  angle  PCK?  of  PC  A 
plus  ACK?  of  PCB  plus  BCK?  of  ACK  plus  KCB?  What 
kind  of  angles  are  they?  If  angle  PC  A  is  equal  to  100°,  what 
is  angle  ACK  equal  to?  Repeat  with  the  various  combina- 
tions, giving  them  different  values. 

When  the  class  has  grasped  the  above,  take  up  the  assign- 
ment. 

The  Assignment.  —  i.   Statement  of  the  problem. 

2.  Deduction  of  its  proof. 

3.  Explanation  of  steps  in  proving  a  geometric  proposition. 

4.  Rules  to  govern  the  study  of  a  proposition. 


ii2    Supervised  Study  in  Mathematics  and  Science 

The  Statement  of  the  Problem.  Explain  that  "  proposition  " 
is  a  general  term  for  either  a  theorem  or  a  problem.  A  theorem 
is  a  geometric  proposition  requiring  proof  ;  a  problem  is  a 
geometric  proposition  requiring  construction.  The  propo- 
sition for  to-day's  consideration  is  in  the  form  of  a  theorem, 
thus: 

Theorem.  —  //  two  straight  lines  intersect,  the  resulting 
vertical  angles  are  equal. 

Deduction  of  the  Proof.  Using  the  figure  already  drawn  by 
the  teacher  and  referred  to  above,  ask  what  vertical  angles 
may  be  considered.  Since  there  are  two  groups,  we  may  study 
either  pair,  or  ACK  and  PCB. 

Ask  someone  to  state  all  the  properties  we  know  about  the 
various  angles  involved  in  this  figure.  Have  a  pupil  write 
these  properties  in  algebraic  form  upon  the  board,  as  : 


II.   PC  'A-}-  ACK  =  180°. 
III. 
IV. 


What  peculiar  thing  is  noticeable  about  these  equations? 
That  they  are  all  equal  to  180°  or  the  same  thing.  What  can 
be  said  about  things  that  are  equal  to  the  same  thing?  That 
they  are  equal  to  each  other.  Let  us  see  if  this  will  be  of  any 
value  to  us. 

What  further  significance  can  we  note  in  the  first  two  equa- 
tions besides  the  fact  that  they  are  both  equal  to  180°?  That 
each  contains  the  same  angle,  PC  A.  What  similar  observa- 
tion can  be  made  with  reference  to  the  second  and  third 
equations?  That  angle  ACK  is  common.  And  in  the 
third  and  fourth  that  the  angle  KCB  is  common.  In  our 


Divisions  of  Plane  Geometry  113 

consideration  of  the  above  proposition,  which  of  these  three 
common  angles  might  be  of  interest  to  us?  The  class  may 
select  ACK.  We  shall  take  the  second  and  third  equations 
containing  this  angle,  and  combine  them  into  a  new  equation 
since  they  are  equal  to  each  other : 

PC  A  +ACK  =  ACK+  KCB. 

Have  someone  state  why  this  is  true.  How  can  algebra  be 
of  service  here? 

But  since  we  have  a  common  term  on  both  sides  of  the  equa- 
tion, what  can  we  do  with  it?  We  can  subtract  this  common 
angle  from  both  sides  without  changing  the  value  of  the 
equation.  Why  ?  Then  we  have : 

PC  A  =  KCB. 

What  kind  'of  angles  are  these?  Then  we  have  proved 
that  the  two  vertical  angles  are  equal.  But  since  these  are 
not  the  two  that  we  started  out  to  prove  equal,  we  will  try 
another  set  of  equations. 

Ask  what  angles  we  want  left  after  solving  the  equation. 
ACK  and  PCB.  Then  let  us  take  two  equations  which  have 
these  angles  and  also  a  common  angle.  The  class  will  readily 
select  the  third  and  fourth,  or 

ACK+  KCB  =KCB+PCB.     (Why?) 
ACK  =  PCB,  (Why?) 

or  what  we  wished  to  prove. 

Now,  draw  two  other  intersecting  lines  upon  the  board, 
and  call  on  someone  to  designate  them  by  other  letters,  to 
find  the  various  equations,  to  pick  out  the  desired  ones  and 
prove  the  proposition.  The  pupil  will  tell  what  to  do  and  the 
teacher  will  write  the  necessary  operations  on  the  board. 


ii4    Supervised  Study  in  Mathematics  and  Science 

Then,  have  all  the  class  draw  two  intersecting  straight  lines 
upon  their  papers,  and  go  through  the  same  process  them- 
selves, calling  you  to  their  aid  if  they  get  into  difficulty.  The 
pupils  will  respond  to  this  with  avidity  because  all  boys  and 
girls  like  to  achieve  things  themselves. 

Next,  have  them  open  their  textbooks,  which  have  been 
closed  up  to  this  time,  and  follow  out  the  similar  proof  there 
developed.  They  will  be  very  pleased  to  learn  that  they  have 
already  mastered  the  new  lesson,  and  they  have  incidentally 
been  given  a  lesson  in  how  to  study. 

An  Explanation  of  the  Steps  to  Take  in  Solving  a  Geometric 
Proposition.  Explain  that  every  demonstration  of  a  geometric 
proposition  is  divided  into  definite,  logical  steps,  as 

a.  Statement  of  the  theorem. 

b.  Drawing  of  the  figure. 

c.  Stating  data  given  in  the  theorem. 

d.  Stating  what  is  given  to  prove. 

e.  The  proof,  consisting  of  steps  and  reasons. 

/.    Conclusion  (Q.E.D.  or  quod  erat  demonstrandum). 

Rules  on  How  to  Study  a  Proposition.  The  following  set  of 
rules  may,  with  excellent  results,  be  mimeographed  and 
distributed  to  the  class,  as  was  suggested  in  algebra. 

SUGGESTIONS  FOR  EFFECTIVE  STUDYING 

Theorem.  —  a.   Read  and  reread  the  theorem  very  carefully. 

b.  Note  what  is  given  and  what  is  wanted. 

c.  Review  in  your  mind  the  properties  of  all  geometric  terms 
occurring  therein. 

d.  Have  a  blank  card,  the  size  of  the  printed  page  of  your 
book ;  call  this  card  No.  i.     Have  another  card  the  length  of  the 
page  but  only  one  half  as  wide ;  call  this  card  No.  2.     With  card 


Divisions  of  Plane  Geometry  115 

No.  i  cover  up  all  the  page  except  the  theorem.  Do  not  uncover 
any  more  of  the  page  until  you  have  mastered  the  above  directions. 

Figure. — e.  On  blank  paper  make  drawings  to  conform  to  the 
data  given  in  the  theorem. 

/.  Push  down  the  large  card  to  disclose  the  drawing  in  the  book 
and  compare  with  yours.  Do  not  put  in  any  auxiliary  lines  until 
later ;  letter  your  figure. 

Data.  —  g.  Write  on  your  paper  under  the  head  Data,  the  things 
you  have  stated  concerning  the  theorem.  Again  push  down  the 
large  card  to  compare.  If  your  statement  of  data  does  not  agree 
with  the  book,  note  wherein  it  differs  and  thoroughly  understand  it 
as  given  by  the  author  before  you  proceed. 

To  Prove. —  h.  Repeat  this  operation  with  the  statement  To 
Prove.  Under  this  heading  is  given  the  thing  desired  by  the  theorem. 
(See  suggestion  b.) 

i.  Divide  the  remainder  of  your  paper  into  two  equal  parts  by 
a  vertical  line.  Label  the  first  column  Steps  and  the  second  column 
Reasons. 

Proof.  — j.  Slip  down  your  leading  card  No.  i  to  uncover  the 
first  step,  keeping  card  No.  2  over  the  corresponding  reason. 

k.  If  this  statement  incurs  auxiliary  lines,  make  them  on  your 
figure  and  compare  for  correctness  with  the  figure  in  the  book. 

/.  Try  to  state  a  reason  why  this  may  be  done.  Then  slip 
down  card  No.  2  to  disclose  the  author's  reason  and  note  whether 
you  were  right.  If  not,  master  the  correct  reasoning  for  the  opera- 
tion before  proceeding. 

m.  Repeat  with  the  next  step  in  the  demonstration,  disclosing 
first  the  step  and  then  the  reason  for  it  after  you  have  attempted  to 
discover  it  for  yourself. 

n.  Repeat  the  process  until  you  have  completed  the  demonstra- 
tion. You  will  then  have  the  complete  work  upon  paper,  also. 

o.  Now  close  your  book  and  after  destroying  your  paper,  try 
to  write  the  complete  demonstration  upon  new  paper.  When 
you  are  unable  to  proceed,  refer  to  your  text,  but  never  memo- 
rize the  steps,  or  you  will  never  really  understand  geometry. 

In  General. — p.  Understand  each  step  absolutely  before  proceed- 
ing any  further. 


n6    Supervised  Study  in  Mathematics  and  Science 

q.  Take  time.  Thoroughly  studying  the  theorem  once  should 
be  enough. 

r.   Always  ask  yourself  why  after  each  step. 

5.   Number  the  steps  and  the  reasons  to  correspond. 

/.  Master  the  work  from  day  to  day ;  do  not  let  it  master  you. 
If  you  rely  on  memorization,  you  will  become  the  slave  instead  of 
the  master  of  this  subject. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 
Proposition  I. 

II  or  Average  Assignment.     Exercises  on  Proposition  I. 

III  or   Maximum   Assignment.     Prove   the   theorem   in- 
formally by  referring  to  Slaught  and  Lennes'  Plane  Geometry,1 

PP-  33>  38,  79,  103- 

The  Silent  Study.  —  Pass  from  pupil  to  pupil  to  see  that 
they  are  following  out  the  directions  as  given  above  for  the 
study  of  the  lesson.  Explain  that  a  little  practice  with  this 
method  will  soon  make  it  automatic,  and  the  pupil  will  find 
that  before  long  he  will  master  a  demonstration  in  a  short 
time. 

Insist  on  carefully  drawn  figures,  neat  and  clearly  lettered. 
Show  the  pupils  how  to  make  the  cards  suggested.  Each 
one  should  be  labeled  and  kept  permanently  in  the  book. 
They  may  be  made  of  paper  if  desired ;  their  value  lies  in 
keeping  from  the  pupil's  observation  all  except  that  which  is 
being  studied  at  the  moment.  Where  they  have  been  used, 
they  have  given  good  results.  Their  use  should  be  encouraged 
by  the  teacher  until  the  pupils  realize  their  value  as  an  aid 
to  study. 

1  Allyn  and  Bacon. 


Divisions  of  Plane  Geometry  117 

LESSON  III 

UNIT   OF   INSTRUCTION   II.  —  Book  I 
RECTILINEAR   FIGURES 

LESSON  TYPE.  —  A  DEDUCTIVE  WESSON 
Program  or  Time  Schedule 

V 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.  Theorem.  If  two  straight 
lines  intersect,  the  vertical  angles  are  equal. 

Method.  Send  two  or  three  pupils,  who  completed  the 
maximum  assignment,  to  the  board  to  write  out  the  demonstra- 
tion of  the  theorem  in  full.  Assign  to  those  completing  the 
average  assignment  the  examples  of  the  text  to  be  worked 
upon  the  board.  Review  the  demonstration  of  the  theorem 
with  the  others. 

Have  the  figure  already  drawn  upon  the  front  board.  Call 
on  someone  to  state  the  theorem,  someone  else  to  give  the  data, 
and  so  on  through  the  proof.  When  the  demonstration  has 
been  completed  in  this  way,  have  someone  stand  and  go  through 
the  whole  demonstration  orally.  After  two  or  three  have  done 
this,  erase  the  figure  and  have  someone  prove  it,  carrying  the 
figure  in  his  mind.  This  is  good  practice  if  carefully  super- 
vised, for  it  keeps  the  minds  of  all  concentrated  on  the  de- 
velopment of  the  proof.  Occasionally  break  in  on  the  demon- 
stration to  call  on  another  to  proceed  with  it.  It  is  a  good 
plan  to  have  the  class  know  that  anyone  is  likely  to  be  called 
upon  at  any  time ;  they  will  then  give  better  attention  to  what 
is  going  on. 

After  the  theorem  has  been  thoroughly  reviewed,  ask  for 


n8    Supervised  Study  in  Mathematics  and  Science 

the  answers  to  the  examples  of  the  assignment,  noting  those 
that  are  correct  upon  the  board.  Also  have  the  pupils  who 
completed  the  maximum  assignment  tell  how  the  theorem 
was  proved  informally  according  to  the  references  given. 
We  are  now  ready  to  take  up  the  new  lesson. 

The  Assignment.  —  i.   Study  of  definitions. 

2.    Explanation  of  the  new  theorem. 

The  Study  of  Definitions  of  Triangles.  Read  the  definitions 
over  with  the  class,  showing  the  pupils  how  to  study  them. 
Tell  the  pupils  that  the  main  points  are  to  master  each  word  as 
they  read  it ;  to  follow  out  all  references  to  figures  ;  to  look  up 
the  meaning  of  all  the  words  they  do  not  know,  and  the  proper 
pronunciation  of  words  they  cannot  pronounce.  Then  take  up 
the  new  proposition. 

Explanation  of  the  New  Theorem.  Follow  the  plan  given  in 
the  preceding  lesson.  As  new  propositions  occur  from  time  to 
time,  much  of  this  work  may  be  shortened  as  the  pupils  are 
able  to  do  more  of  the  deductive  work  themselves,  following 
out  the  suggestions  of  the  preceding  rules.  The  teacher 
should  determine,  from  day  to  day,  what  will  probably  con- 
stitute the  real  difficulties  and  should  clear  these  up  during 
the  assignment  period.  It  is  better,  however,  to  leave  as 
much  as  possible  to  the  silent  study  period  and  give  those 
capable  of  solving  the  problems  unaided  a  chance  to  do  so. 
The  class  demonstration  assumes  that  all  are  equally  unable  to 
study  it  out  for  themselves,  which  is  not  only  not  a  fact  but  is 
stultifying  to  the  more  capable  pupils. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 
Proposition  II  on  triangles. 

//  or  Average  Assignment.  Simple  exercises  on  Proposi- 
tion II. 


Divisions  of  Plane  Geometry  119 

777  or  Maximum  Assignment.  Examples  on  page  28, 
Wentworth  and  Smith's  Plane  Geometry.1 

The  Silent  Study  Period.  —  The  above  assignment  having 
been  placed  in  advance  upon  the  board,  the  pupils  are  ready 
to  begin  their  silent  study  of  the  lesson.  Tell  them  to  open 
their  books  at  the  new  proposition,  and  to  study  it  according 
to  the  directions  given  them  yesterday.  With  the  large  card 
over  all  the  page  except  the  theorem,  they  should  begin  with 
rule  a.  For  a  day  or  two,  or  until  they  have  acquired  the 
correct  method  of  studying  the  demonstration  as  given  in  the 
book,  it  will  be  best  for  the  teacher  to  direct  orally  this 
study,  guiding  the  pupils  to  apply  correctly  these  direc- 
tions. Since  one  purpose  of  supervised  study  is  to  develop 
eventually  in  the  pupil  a  knowledge  of  how  to  study, 
too  much  emphasis  cannot  be  placed  upon  this  practice. 
In  time,  if  such  study  is  directed  and  insisted  upon,  the  pupils 
will  find  it  unnecessary  to  rely  upon  the  teacher's  guidance 
and  will  be  able  to  study  the  lesson  unaided.  This  should 
be,  of  course,  the  ideal  sought  for,  but  it  will  take  more 
or  less  time  and  the  acquisition  will  only  come  through 
patience  and  perseverance. 

After  the  pupils  have  been  given  time  to  digest  the  stated 
theorem,  ask  someone  to  state  what  is  given  and  what  is 
wanted.  Not  until  it  is  clearly  understood  that  two  triangles 
are  given,  each  of  which  has  two  sides  of  one  equal  to  two  sides 
of  the  other,  and  the  included  angle  of  the  one  equal  to  the 
corresponding  included  angle  of  the  other,  and  that  we  desire 
to  prove  that  under  these  conditions  the  two  triangles  are 
equal,  are  we  ready  to  pass  to  the  third  rule.  Call  on  various 
pupils  to  state  these  things.  A  clear  comprehension  of  these 

1  Ginn  and  Co. 


i2o    Supervised  Study  in  Mathematics  and  Science 

two  elements  of  every  proposition  is  absolutely  necessary 
before  we  proceed. 

Now  ask  what  we  know  concerning  the  properties  of  the 
geometric  terms  involved  in  this  proposition.  Bring  out  that 
all  triangles  have  three  sides ;  that  there  are  also  three  angles 
and  that  each  angle  is  included  between  two  sides.  Such  an 
angle  is  called  an  included  angle.  Why?  Ask  when  geo- 
metric figures  may  be  said  to  be  equal.  If  the  pupils  are 
unable  to  tell,  refer  them  to  the  axiom.  Having  now  analyzed 
all  of  the  properties,  turn  to  rule  d.  While  they  keep  the  large 
card  over  all  of  the  page  except  the  statement  of  the  propo- 
sition, they  will  next  draw  figures  upon  their  papers  to  con- 
form to  the  facts  of  the  theorem.  When  this  has  been  done, 
tell  them  to  lower  the  card  to  disclose  the  figure  and  to  compare, 
lettering  it  to  conform  with  the  lettering  of  the  drawing  in  the 
textbook. 

Next,  tell  them  to  write  after  the  heading  Given  the  facts 
that  have  been  stated  in  the  theorem  as  data.  Before  the 
card  is  pushed  down  again,  ask  someone  to  tell  you  what  he 
has  written  under  this  head.  Discuss  it  with  the  class  to  see 
if  it  is  complete.  Then  tell  them  to  uncover  this  section  of  the 
book  to  compare  with  the  statement  of  the  author. 

Repeat  the  process  with  the  heading  To  Prove.  In  order 
to  expedite  the  work,  the  teacher  may  discuss  each  of  these 
points  orally  rather  than  have  the  pupils  write  them  down, 
possibly  incorrectly.  The  thing  desired  is  the  deduction  of 
the  various  steps,  as  far  as  possible,  by  the  pupils  themselves 
without  referring  to  the  book,  except  for  verification. 

The  same  procedure  may  be  followed  throughout  the  study 
of  the  demonstration.  The  use  of  the  divided  card,  as  ex- 
plained in  Lesson  II,  will  tend  to  make  the  pupil  study  each 


Divisions  of  Plane  Geometry  121 

step  out  himself  before  referring  to  the  author.  This  should 
especially  be  insisted  upon  in  giving  the  reasons. 

After  the  first  step  in  the  proof  has  been  studied,  ask  the 
pupils,  preferably  individually,  to  give  some  reason  why  this 
step  has  been  taken.  The  teacher  will  thus  get  all  the  class  to 
thinking,  and  they  will  review  mentally  all  the  facts  which  they 
have  learned  up  to  this  time.  This  is  a  point  which  cannot  be 
overemphasized.  Ability  to  solve  originals  or  in  fact  to  do  any 
part  of  the  work  in  geometry  requires  a  continual  revolving 
in  the  mind  of  all  our  previous  knowledge  with  a  view  of 
applying  it  to  the  specific  case  under  consideration.  The 
difficulty  with  originals  often  results  from  this  very  inability 
or  failure  to  practice. 

When  the  complete  demonstration  has  been  studied  in  this 
way,  the  teacher  may  tell  the  class  to  take  a  sheet  of  paper  and 
try  to  rewrite  the  proof,  or  better,  to  reconstruct  the  proof. 
The  only  thing  which  will  at  all  necessitate  the  use  of  the 
memory  will  be  the  order  of  the  steps  and  the  continual  review 
mentally  of  previously  learned  facts  concerning  geometric 
terms. 

As  the  pupils  begin  this  work,  the  teacher  will  pass  among 
them  to  note  any  special  difficulties.  Occasionally  a  leading 
question  will  set  the  pupil  aright  but  definite  answers  should 
be  avoided ;  it  is  our  duty  to  lead  the  pupil  to  see  his  trouble 
rather  than  simply  to  find  it  for  him. 

The  Study  of  the  Advance  Assignments.  —  As  soon  as  a 
pupil  has  mastered  the  new  proposition,  he  should  go  to  work 
upon  the  second  assignment.  It  will  be  best  at  first  to  demand 
that  the  answers  to  these  exercises  be  written  out.  It  will 
take  more  time  and  later  may  be  dispensed  with,  but  at  first 
it  will  cause  the  pupil  to  concentrate  his  attention  more 


122    Supervised  Study  in  Mathematics  and  Science 

definitely  upon  his  work.  These  first  exercises  are  simple  and 
will  not  take  much  time  or  paper,  but  the  pupil  by  this  method 
will  soon  acquire  the  habit  of  being  accurate,  comprehensive, 
and  neat. 

In  case  there  are  any  exercises  that  might  involve  special 
difficulty,  a  leading  question  concerning  it  may  be  written 
upon  the  board,  numbered  to  refer  definitely  to  such 
exercises. 

The  examples  assigned  for  the  maximum  assignment  may 
either  be  written  upon  the  board  or  the  pupil  allowed  to  take 
the  book  referred  to.  In  case  more  than  one  should  reach 
these  examples  during  the  period,  the  difficulty  arising  from 
having  only  one  copy  would  be  eliminated  by  the  use  of  the 
board.  On  the  other  hand,  the  actual  use  of  supplementary 
textbooks  by  the  pupil  is  an  excellent  practice  and,  whenever 
it  is  feasible,  such  additional  books  should  be  provided  and 
in  as  large  a  variety  as  possible. 

Encourage  the  pupils  who  are  capable  to  attempt  the 
maximum  assignment  but  not  to  the  detriment  of  their  other 
work.  Make  it  clear,  too,  that  it  is  only  to  be  studied  after 
the  mastery  of  the  first  and  second  parts.  Its  purpose  is  to 
keep  employed  those  in  the  class  who  are  especially  capable  and 
thus  further  to  develop  their  powers.  As  a  rule,  these  ex- 
amples should  be  of  a  more  difficult  nature.  This  method, 
carried  out  to  its  ideal  administration,  would  permit  these 
pupils  to  proceed  with  new  propositions,  and  so  finish  the 
book  ahead  of  the  others  ;  but  this  involves  too  much  difficulty 
for  the  average  school  to  attempt  and  should  only  be  done 
under  exceptional  circumstances.  For  a  further  discussion 
of  this  important  phase  of  supervised  study  as  relating  to 
higher  mathematics,  see  Chapter  Four. 


Divisions  of  Plane  Geometry  123 

LESSON  IV 

UNIT  OF  INSTRUCTION  H.  —  Book  I 
RECTILINEAR  FIGURES 

LESSON  TYPE.  —  A  How  TO  STUDY  LESSON 
Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.  The  first  three  propositions 
of  Book  I. 

Method.  Send  three  pupils  to  the  board  to  draw  the  figures 
of  the  propositions  which  are  to  be  reviewed.  While  these  are 
being  put  on  the  board,  quickly  review  the  class  on  the  leading 
facts  already  studied  in  regard  to  angles,  triangles,  etc.  Such 
questions  as  the  following  are  suggestive : 

What  kind  of  angles  are  equal  ? 

State  under  what  two  conditions,  already  studied,  triangles  are 
equal. 

What  method  of  proof  is  employed  to  prove  triangles  equal  ? 
When  is  a  line  perpendicular  to  another  line  ? 
What  do  you  mean  by  bisecting  a  line  f    an  angle  ? 

The  questions  asked  should  always  review  the  knowledge 
acquired  recently,  and  facts  that  might  be  useful  in  studying 
new  propositions  and  originals  should  be  gradually  introduced 
by  this  means. 

The  figures  having  been  drawn  upon  the  board,  call  on  some 
pupil  to  state  the  theorem  about  vertical  angles  and  to  prove 
it,  using  the  figure  on  the  board.  See  that  he  omits  nothing 
important.  It  is  best  to  prompt  the  one  reciting  as  little  as 
possible,  because,  if  the  pupil  learns  to  depend  on  the  teacher 


124    Supervised  Study  in  Mathematics  and  Science 

for  his  approval  or  disapproval  of  every  statement,  he  will 
never  be  able  alone  to  complete  a  demonstration.  The  teacher 
must  avoid  aiding  him  too  much ;  if  the  instructor  breaks  in 
with  "  why  "  every  time  the  pupil  fails  to  give  a  reason,  he 
will  early  learn  to  expect  it  and  will  not  trouble  himself  to  give 
it  unaided.  A  better  way  is  to  stop  him  when  he  makes  an  error 
and  let  someone  else  proceed  with  the  proof.  He  will  soon 
learn  that  he  must  rely  on  himself,  and  will  exert  himself  to 
be  thorough.  After  another  pupil  has  completed  the  demon- 
stration successfully,  tell  the  one  who  failed,  unless  he  himself 
senses  it,  wherein  he  made  his  mistake,  and  let  him  try  again. 

Proved  orally,  many  demonstrations  can  be  covered  in  the 
allotted  time.  As  soon  as  this  proposition  is  given  correctly 
and  the  pupil  who  failed  has  been  able  without  help  to  give 
it,  call  on  someone  to  prove  the  second  proposition,  and  so  on 
until  the  subject  matter  of  the  review  has  been  covered.  It  is 
well  to  review  each  day  not  only  the  propositions  which  were 
the  immediate  subject  of  study  for  that  day,  but  also  to  review 
continually  others,  dwelling  especially  on  those  which  have 
given  the  most  difficulty.  In  cases  of  complicated  figures,  let 
the  pupil  go  to  the  board  and  use  the  pointer,  and  in  the  case 
of  many  equations  being  necessary,  it  will  help  the  pupil  to 
allow  him  to  write  them  on  the  board. 

If  there  are  some  in  the  class  who  still  have  difficulty  and  are 
unable  to  go  through  a  demonstration  correctly,  entirely  alone, 
let  them  go  to  the  board  during  the  study  period  and  write  it 
out.  This  method,  however,  although  much  used,  is  of  special 
value  only  in  the  case  of  pupils  who  are  unable  to  prove  the 
proposition  orally  before  the  class.  It  ordinarily  takes  too 
much  time  to  be  of  much  worth. 

When  a  pupil  is  demonstrating  a  proposition,  the  teacher 


Divisions  of  Plane  Geometry  125 

must  bear  in  mind  that  if  the  time  thus  spent  is  not  to  be 
wasted  by  the  rest  of  the  class,  the  pupil  must  talk  loud  enough 
for  all  to  hear  him  clearly.  Let  him  stand  at  one  side  of  the 
figure  and  talk  to  the  class.  The  pupil  should  be  made  to  feel 
that  he  is  taking  their  time  and  they  are  entitled  to  all  the  bene- 
fits that  may  accrue  to  them  from  his  work. 

A  few  minutes  might  also  be  spent  in  running  over  the 
exercises,  which  are  based  on  the  day's  lesson.  These  may  be 
given  orally  or  written  upon  the  board,  depending  on  their 
nature  and  their  value.  When  possible,  similar  exercises  with 
different  values  may  well  be  given  by  the  teacher ;  and  the 
pupils  may  be  encouraged  to  make  up  others  of  a  like  nature. 

The  Assignment.  —  The  new  lesson  will  be  on  originals ;  and 
since  it  is  the  first  lesson  on  these,  a  few  words  as  to  how  to 
study  them  properly  will  not  be  out  of  place. 

Notebooks.  The  author  advises  the  use  of  notebooks ;  one 
to  use  in  class  for  original  exercises  and  to  be  handed  over  to  the 
teacher  at  the  close  of  the  period ;  the  other  for  use  outside  of 
class  for  additional  exercises  which  were  not  done  under  the 
immediate  observation  of  the  teacher.  The  former  will 
contain  beyond  peradventure  the  pupil's  own  work ;  the  latter 
may  be  assumed  to  be  such  and  assessed  according  to  the 
ability  of  the  pupil  to  work  similar  examples  in  class. 

The  pupils  are  told  to  open  their  textbooks  and  to  read  the 
first  exercise.  Opening  their  notebooks,  they  will  draw  the 
figure,  state  what  is  given,  and  what  is  to  be  found.  Ask 
various  questions,  taking  pains  to  see  that  all  understand  the 
data  given  and  the  results  desired.  The  questions  that  you 
will  ask  to-day  will  illustrate  how  the  pupil  is  later  to  question 
himself  when  studying  similar  problems.  The  following 
illustrations  may  serve  to  make  the  method  clear. 


126    Supervised  Study  in  Mathematics  and  Science 

Example.  Given  lines  AB  and  CD  bisecting  each  other  at  O. 
Draw  straight  lines  connecting  A  and  C,  and  D  and  B. 
Prove  that  AACO  =  AOBD. 

Questions.  When  lines  bisect  each  other,  what  results  ?  Which 
segments,  then,  are  made  equal  in  this  example  ? 

To  prove  that  triangles  are  equal,  we  must  try  them  by  three 
conditions ;  what  are  they  ? 

In  this  figure,  what  do  we  know  about  each  triangle  ? 

Does  this  give  us  enough  knowledge  of  each  to  throw  them  into 
one  of  the  conditions  under  which  triangles  may  be  proved  equal  ? 

Then,  since  we  know  that  each  triangle  has  two  sides  and  the 
included  angle  equal  respectively  to  two  sides  and  the  included 
angle  of  the  other,  what  must  follow? 

Is  this  what  we  wish  to  prove  ? 

Solution.  Tell  the  pupils  to  write  out  later  the  entire  proof 
according  to  the  method  of  the  demonstrations  in  the  textbook, 
using  the  divided  page,  with  the  steps  on  the  left  of  the  line  and 
the  reasons  on  the  right. 

Example.  Given  Z  ABC  bisected  by  BY,  P  is  any  point  in  BY, 
equal  lines  are  dropped  from  P  to  the  sides  of  the  angle,  as  PM 
and  PK.  Prove  that  ABPM  =  ABPK. 

Questions.  First  ask  the  pupils  what  questions  they  think  they 
should  ask  themselves  concerning  this  example.  Naturally  many 
questions  will  come  to  the  mind  of  the  individual  pupil  which  will 
be  found  to  have  no  bearing  on  the  problem  under  consideration, 
but  it  will  be  best  to  exhaust  all  possible  conditions  even  at  the 
risk  of  some  matter  which  is  not  pertinent.  He  will  learn  that  in 
the  investigation  of  any  new  subject,  much  of  the  effort  exerted 
fails  to  be  of  consequence,  but  it  is  necessary  to  bring  to  bear  all 
the  known  facts  so  that  we  may  study  their  relationships. 

Explain  that,  in  the  first  place,  the  pupils  must  study  the 
examples  by  passing  over  in  their  minds  all  possible  related 
facts,  and,  in  the  second  place,  by  eliminating  all  except  those 
that  will  bear  directly  on  the  problem  involved.  We  know 
such  and  such  a  thing.  What  ought  we  to  know  if  we  solve 


Divisions  of  Plane  Geometry  127 

the  problem?  Do  these  facts  help  us  and  how?  If  not,  are 
there  any  others  that  we  have  omitted  that  we  could  possibly 
use? 

In  this  way  analyze  five  or  six  examples  in  to-morrow's 
assignment,  and  then  require  the  pupils  to  write  out  in  full  in 
their  notebooks  the  demonstration  of  each  in  the  way  pre- 
viously indicated. 

The  Value  of  Definite  Rules  for  the  Study  of  Different  Phases 
of  the  Subject.  Give  each  pupil,  on  mimeographed  sheets  if 
possible,  the  following  rules  for  attacking  and  solving  original 
exercises.  If  he  will  adhere  rigidly  and  conscientiously  to  these 
directions  he  will  be  able  to  solve  any  original  exercise  he  may 
meet.  And  not  only  that,  he  will  save  himself  the  time  that 
is  often  spent  in  aimless  study  —  study  that  brings  to  bear  no 
intelligent  and  directed  effort.  It  is  comparable  to  the  way 
the  expert  machinist  looks  for  "  trouble  "  and  the  way  the 
unskilled  layman  looks  for  it.  The  expert  does  not  aimlessly 
take  off  nuts,  loosen  joints,  and  dislocate  couplings ;  before 
he  touches  a  thing,  he  studies  his  problem.  How  should  the 
machine  function;  and,  is  it  so  functioning?  If  not,  what 
might  cause  such  trouble?  And  so  on,  until  he  finally  elimi- 
nates many  improbable  or  impossible  causes  and  reduces  it  to 
something  which  is  probable  or  possible,  and  then  he  goes 
after  that  thing.  The  layman,  on  the  other  hand,  not  studying 
it  out  beforehand,  will  do  something  here  and  something  there, 
until  the  chances  are  he  will  add  to  his  trouble  instead  of  re- 
moving it.  So  in  solving  exercises  in  geometry,  a  little  care- 
ful analyzing  of  the  data,  the  problems  involved,  the  things 
known  and  the  things  desired,  and  how  one  may  affect  the 
other,  will  lead  eventually  and  logically  to  the  correct 
solution. 


128    Supervised  Study  in  Mathematics  and  Science 

SUGGESTIONS  FOR  STUDYING  ORIGINALS 

a.  Digest  every  word  in  the  problem. 

b.  Make  the  figure  carefully  and  go  over  it  to  see  that  it  follows 
directions. 

c.  Ask  yourself  what  you  know  about  every  word  in  the  data. 

d.  Ask  yourself  how  this  knowledge  may  be  applied  to   the 
question  under  consideration. 

e.  State  carefully  what  is  to  be  proved. 

/.   Review  mentally  under  what  conditions  the  proof  may  be  made. 
g.   Write  out  the  proof  in  full,  giving  a  reason  for  each  step. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 
Exercises  1-6,  in  text. 

//  or  Average  Assignment.     Exercises  7-9,  in  text. 

///  or  Maximum  Assignment.     Exercises  10  and  u,  in  text. 

Method  of  Manipulating  the  Notebooks.  As  noted  above, 
have  all  the  pupils  write  out  in  their  notebook,  or  Notebook  A , 
all  the  exercises  worked  during  the  class  period ;  work  in  the 
other  notebook,  or  Notebook  B,  those  done  outside  of  class. 
At  the  close  of  the  period,  stand  at  the  door  and  collect  the 
former  notebooks.  It  will  take  only  a  few  minutes  to  look 
them  over  and  check  those  found  to  be  incorrect.  Make  a 
note  of  these  failing  pupils  on  the  back  of  the  day's  assign- 
ment sheet,  and  take  up  the  unsolved  problems  with  these 
pupils  the  next  day. 

The  notebook  is  easily  kept  in  order,  it  indicates  the  pupil's 
work  and  progress  from  day  to  day  and  becomes  an  efficient 
reminder  of  poor  work.  Such  remarks  as  "  too  slovenly," 
"  don't  guess,"  "  figures  inaccurate,"  etc.,  might  well  call  the 
attention  of  the  pupil  to  the  reason  for  his  failures.  Rubber 
stamps  with  such  phrases  could  be  used  to  advantage  with  a 
view  of  saving  time  and  energy.  Written  work  taken  up  but 
never  returned  with  criticism  is  inefficient  and  does  not  repay 


Divisions  of  Plane  Geometry  129 

the  teacher  for  the  time  he  spends  looking  it  over.  Papers 
handed  in  and  simply  marked  with  a  grade  or  per  cent  are  of 
little  help  to  teacher  or  pupil.  The  real  benefits  are  derived 
only  when  the  errors  are  noted  and  the  attention  of  the  pupil 
called  to  them  so  that  he  may  avoid  making  similar  mistakes. 

The  notebooks  containing  the  work  done  outside  of  the 
class  period  may  be  looked  over  during  the  study  of  the  assign- 
ment and  errors  brought  at  once  to  the  notice  of  the  pupil. 
They  may  be  returned  to  the  pupils  at  the  close  of  the  period, 
or  collected  only  as  the  teacher  is  able  to  look  them  over. 
Hence  the  value  of  notebooks ;  they  are  always  ready  for 
inspection,  and  their  administration  is  very  easily  effected. 

All  pupils  ought  not  to  be  required  to  do  the  same  amount 
of  original  work ;  all,  however,  should  do  the  minimum  assign- 
ment, which  should  cover  all  the  simpler  applications.  The 
more  efficient  workers  should  be  encouraged  to  solve  more 
difficult  exercises,  and  so  all  may  be  kept  up  to  the  limit  of 
their  respective  capacities. 

LESSON  V 

UNIT   OF  INSTRUCTION    H.  —  Book  I 
RECTILINEAR   FIGURES 

LESSON  TYPE.  —  A  DEDUCTIVE  LESSON 
Program  or  Time  Schedule 

The  Review 30  minutes 

The  Assignment 5  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.     Original  exercises. 
Method.    The  figures  for  the  exercises  which  were  assigned 
in  yesterday's  lesson  and  which  are  to  be  reviewed  to-day 


130    Supervised  Study  in  Mathematics  and  Science 

should  be  placed  upon  the  blackboard  prior  to  the  assembling 
of  the  class.  By  a  survey  of  the  A  notebooks,  handed  in 
at  the  close  of  the  period  the  day  previous,  the  teacher  will 
have  noted  those  examples  which  have  given  trouble.  This 
may  be  easily  kept  in  mind  by  a  simple  method,  that  of 
noting  on  the  back  of  the  assignment  sheet,  or  any  piece  of 
paper,  the  examples  by  number  and  beside  the  numbers  the 
names  of  the  pupils  having  difficulty  with  them.  The  exact 
trouble  might  also  be  indicated.  These  memoranda  might 
read  as  follows : 

No.  i.  O.K. 

No.  2.   O.K. 

No.  3.  John  Jones  (does  not  understand  meaning  of  midpoint). 

No.  4.   O.K. 

No.  5.   Ben  Ayers. 

No.  6.  John  Jones. 

Ethel  Clare. 

Mollie  Pond. 

Assuming,  then,  that  all  the  pupils  solved  the  first  two 
examples  correctly,  and  that  one  pupil  had  trouble  with 
the  third,  begin  with  that  one.  When  all  have  solved  a 
certain  exercise,  there  is  no  object  in  giving  it  any  more  at- 
tention. 

Ask  John  to  rise  and  read  the  third  example,  which  is  as 
follows : 

Given  0,  the  midpoint  of  the  line  AB,  CO  ±AB,  P  any  point  in 
CO.  Prove  AAPO=AOPB. 

Then  tell  John  to  close  his  book  and  tell  you  the  example  in 
his  own  words.  Until  he  can  do  this  he  is  unable  intelligently 
to  consider  its  solution.  Glancing  at  your  notes,  you  will  see 
that  he  seemed  not  to  sense  the  meaning  of  the  word  "  mid- 


Divisions  of  Plane  Geometry  131 

point."  After  he  has  correctly  stated  the  example,  ask  him 
what  is  given.  Then  ask  him  what  he  understands  by  the 
word  "  midpoint "  and  upon  his  failure  to  tell  you,  call  on 
someone  else  to  explain  its  meaning  and  significance  in  con- 
nection with  this  example.  When  John  realizes  that  0  divides 
the  line  A  B  into  two  equal  parts  or  segments,  he  may  be  able 
to  proceed.  If  not,  direct  him  by  skillful  questioning  until  he 
understands  just  wherein  the  key  to  the  situation  lies.  It  is 
probably  in  the  fact  that  CO  being  perpendicular  to  AB  makes 
equal  angles  at  O  because  they  are  right  angles,  and  all  right 
angles  are  equal.  John  is  not  told  this,  of  course,  but  it  is 
brought  out  either  through  questioning  or  is  answered  by  some 
other  pupil.  Thus  by  guiding  him  to  see  significant  facts 
concerning  the  example,  which  he  failed  to  see  by  himself, 
he  is  being  taught  how  to  study  and  will  feel  within  him- 
self a  growing  power  which  will  stimulate  him  to  greater 
efforts. 

Now  that  he  has  been  guided  through  the  solution,  tell  him 
to  start  over  and  go  through  it  again.  This  may  take  longer 
than  the  teacher  feels  ought  to  be  devoted  to  one  pupil,  but  it 
should  be  of  value  to  the  rest  of  the  class  at  the  same  time, 
and  that  a  few  are  directed  to  find  their  difficulties  and  solve 
the  problem  that  gave  them  trouble  is  worth  much  more 
than  many  exercises  hastily  done  and  nothing  positive 
accomplished.  It  is  not  necessarily  the  number  of  exercises 
that  one  does  which  counts  but  the  ability  to  do  them  by  one's 
self.  It  is  of  more  value  to  a  class  to  conquer  a  few  problems, 
even  if  done  painstakingly  and  with  toil,  than  it  is  to  work 
many  which  require  practically  no  effort.  The  teacher  may 
feel  well  repaid  if  each  day  he  can  help  a  small  number  to 
make  a  definite  advance  in  the  mastery  of  the  subject. 


132    Supervised  Study  in  Mathematics  and  Science 

Repeat  the  above  procedure  with  the  fifth  exercise,  the  next 
one  to  give  trouble,  always  giving  the  problem  or  proposition 
to  the  pupil  who  experienced  difficulty  with  it.  Those  who 
had  no  difficulty  may  at  the  same  time  feel  that  they  also  are 
advancing  for  they  have  the  assurance  that,  through  their 
additional  work  with  the  maximum  assignments  and  their 
work  as  shown  by  the  B  notebooks,  they  are  acquiring  a  real 
grasp  of  the  subject. 

When  the  examples  giving  trouble  have  thus  been  disposed 
of,  if  time  remains,  a  few  similar  examples  from  other  texts 
may  be  given  to  those  who  successfully  completed  the  mini- 
mum and  average  assignments.  If  there  are  some  who  have 
in  addition  completed  part  or  all  of  the  maximum  assignment, 
they  might  be  given  one  of  special  difficulty  and  either  sent  to 
the  board  or  told  to  work  at  their  seats.  Excellent  supple- 
mentary material  may  be  found  in  Schultze  and  Sevenoak's 
Plane  Geometry.1  Special  attention  is  called  to  the  practical 
applications  of  geometry  in  the  back  of  this  book. 

The  Assignment.  —  i.   Study  of  the  notebooks. 

2.   Explanation  of  the  assignment. 

Directions  for  Study  of  the  Returned  Notebooks.  Return 
the  A  notebooks  to  the  class  and  explain  that  you  have 
checked  the  exercises  which  have  been  done  correctly  with  a 
"  C  "  and  that  you  have  marked  errors  made  in  those  done 
incorrectly,  calling  their  attention  to  the  thing  that  caused 
them  trouble.  Tell  them  to  look  these  over  first  and  to  note 
your  remarks.  If  they  do  not  understand  the  remarks,  remind 
them  that  they  may  call  you  to  them  and  you  will  explain  in 
more  detail.  If  their  work  shows  no  errors,  they  may  proceed 
at  once  with  the  advance  assignment. 

1  The  Macmillan  Company. 


Divisions  of  Plane  Geometry  133 

Explanation  of  the  Nature  of  the  New  Assignment  of 
Exercises.  Call  the  attention  of  the  class  to  their  new  assign- 
ment upon  the  board.  Explain  that  the  exercises  are  similar 
to  those  of  to-day,  but  that  some  lack  the  figures.  Call  their 
attention  also  to  the  rules  you  gave  them  yesterday  for  study- 
ing originals  and  explain  that  these  should  be  applied  to  the 
study  of  each  item  in  the  assignment. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 
Exercises  94,  95,  96  (with  figure),  97,  98,  Schultze  and  Seven- 
oak's  Plane  Geometry,  page  26. 

II  or  Average  Assignment.  Exercises  99,  100,  101,  102 
(same  source). 

///  or  Maximum  Assignment.  Exercises  108,  112,  and  115 
(same  source). 

The  Silent  Study.  —  As  soon  as  all  are  at  work  on  the  new 
lesson,  look  over  as  many  of  the  outside  or  B  notebooks  as 
possible,  making  needed  criticisms  as  before.  These  errors 
may  be  called  to  the  attention  of  the  pupils  at  once,  or  at  the 
close  of  the  period.  The  amount  of  time  that  the  teacher  will 
be  able  to  devote  to  this  will  depend  on  the  amount  of  time  he 
will  be  called  upon  to  give  to  pupils  with  their  new  work.  But 
since  the  pupils  will  resume  the  study  of  propositions  to-mor- 
row, it  is  not  necessary  to  review  all  the  work  at  this  time. 
The  work  may  be  collected  to-morrow  and  checked  as  soon 
as  convenient.  Sometimes  the  B  notebook  may  not  be 
required  and  the  work  done  outside  of  class  simply  handed 
in  on  loose  paper.  This  may  be  found  to  be  a  more  satisfac- 
tory way  under  some  conditions,  but  the  notebook  that 
includes  the  work  done  in  class  is  felt  by  the  writer  to  be  very 
important  and  the  work  so  preserved  is  always  available  for 
purposes  of  comparison  or  review. 


134    Supervised  Study  in  Mathematics  and  Science 
LESSON  VI 

UNIT   OF  INSTRUCTION   H.  —  Book   I 

RECTILINEAR  FIGURES 
LESSON  TYPE.  —  SOCIALIZED  REVIEW  LESSON 

Program  or  Time  Schedule 
The  Review 60  minutes 

The  Review.  —  Subject  Matter.    All  propositions  of  Book  I. 

Method.  Have  the  various  theorems  of  Book  I  written 
upon  small  cards  about  two  by  three  inches  in  size.  Shuffle 
these  and,  after  sending  as  many  pupils  to  the  board  as  you 
have  panels,  let  each  one  draw  a  card  and  proceed  to  draw  the 
figure  and  write  out  the  demonstration.  After  the  pupils 
have  worked  awhile,  send  the  one  at  the  extreme  left  to  his 
seat  and  request  the  others  to  move  one  panel  to  the  left,  and 
to  go  on  with  the  demonstration  where  the  other  left  off. 
A  new  pupil  is  sent  to  take  the  place  of  the  one  seated  and  to 
work  the  proposition  at  the  extreme  right.  This  process  may 
be  repeated  at  intervals  of  a  few  minutes  or  done  only  once, 
depending  on  the  judgment  of  the  teacher.  When  an  exercise 
is  completed,  send  another  pupil  with  a  new  card  to  the  board 
to  start  a  new  proposition. 

This  method  of  reviewing  a  number  of  propositions  will 
entail  considerable  work  on  the  part  of  the  teacher,  but  by  care- 
ful planning  it  may  be  very  readily  administered.  Pupils  at 
their  seats  should  in  the  meantime  have  been  given  the  same 
kind  of  work  to  do  on  paper,  which  may  or  may  not  be  inter- 
changed from  time  to  time. 

Some  Advantages  of  This  Kind  of  Reviews  Are: 

1.  All  are  working. 

2.  Each  pupil  is  reviewed  on  a  number  of  propositions. 


Divisions  of  Plane  Geometry  135 

3.  The  pupil's  work  is  under  the  scrutiny  of  teacher  and 
pupil. 

4.  It  emphasizes  the  necessity  for  neatness,  thoroughness, 
and  legibility. 

5.  It  develops  judgment  on  the  part  of  the  pupil  who  takes 
up  the  unfinished  work  as  to  the  correctness  of  that  already 
done. 

6.  It  socializes  the  work  of  all  the  class ;   each  one  is  de- 
pendent not  only  on  what  he  has  done  but  on  what  the  pre- 
ceding pupil  has  done.     Many  modifications  will  occur  to 
different  teachers.     For  instance,  it  may  be  better  to  have 
the  pupils  finish  their  demonstrations  and  then  assign  others 
to  look  them  over  and  report ;  if  any  are  incorrect,  have  these 
erased  and  reworked  by  the  pupil,  or  he  may  be  directed  to 
make  the  corrections  himself. 

Again,  two  pupils  may  be  assigned  to  each  proposition,  one 
to  write  down  the  steps  and  the  other  to  write  down  the  cor- 
responding reasons  in  the  second  column.  These  two  may 
change  places  on  a  given  signal. 

LESSON  VII 

AN  EXHIBITION   OR  RED   LETTER  DAY  LESSON 

Object.     The  object  of  a  lesson  of  this  nature  is  threefold : 

1.  To  bring  out  the  practical  nature  of  geometry. 

2.  To  arouse  in  the  class  the  desire  to  do  superior  work. 

3.  To  further  interest  in  the  subject. 

Place.  This  type  of  lesson,  although  not  to  be  overempha- 
sized, may  with  profit  be  given  at  the  close  of  the  work  on 
each  book. 

Preparation.    Like  all  stated  programs,  the  lesson  should  be 


136    Supervised  Study  in  Mathematics  and  Science 

planned  a  week  or  more  in  advance,  the  different  assignments 
carefully  made,  and  the  purpose  well  explained,  and  it  should 
be  carried  out  according  to  plan. 
Method.    The  program  may  consist  of  three  parts : 

1.  Work  to  be  put  on  display. 

2.  Papers  and  oral  proofs. 

3.  An  exposition  of  some  particular  phase  by  the  teacher, 
principal,  or  superintendent. 

Explain  early  in  the  study  of  the  subject  that  you  are  going 
to  have  an  exhibition  at  the  completion  of  this  book  of  the 
best  work  that  has  been  handed  in  during  the  period  covered. 
This  may  consist  of  the  best  notebooks,  best  construction 
problems,  best  designs,  etc.  Tell  the  pupils  that  from  time  to 
time  you  will  select  the  best  drawings  for  special  attention, 
such  as  having  them  traced  in  India  ink  on  high-grade  paper. 
Encourage  the  pupils  to  do  their  work  from  day  to  day  with 
this  exhibition  in  mind. 

Assign  someone  to  prepare  a  paper  on  some  geometrician's 
life  or  some  interesting  phase  in  the  history  of  the  subject,  or 
some  equally  interesting  topic.  Naturally  the  display  work 
will  be  done  by  the  best  pupils  but  encourage  all  to  do  their 
best  to  get  on  the  program,  and  emphasize  this  as  being  an 
honor. 

The  third  part  of  the  program  will  be  taken  care  of  by  the 
teacher. 

Program.  Keep  the  day  of  the  special  program  before  the 
class  by  skillful  advertising,  so  that  all  will  be  on  the  qui  vive 
for  its  approach.  The  teacher  has  had  it  in  mind  for  some 
time  and  will  have  formed  a  clear  outline  of  just  what  he 
wants  done.  With  careful  planning,  the  following  program 
will  be  found  of  interest  and  at  the  same  time  instructive : 


Divisions  of  Plane  Geometry  137 

PART  I.    DISPLAY 

1.  Two  or  three  of  the  best  notebooks. 

2.  A  few  of  the  best  construction  problems,  drawn  on  high-grade 
drawing  paper  and  traced  in  India  ink. 

3.  A  few  geometric  designs,1  done  by  maximum  pupils  and 
traced  in  colored  inks. 

PART  II.    PAPERS  AND  ORAL  PROOFS 

4.  Paper  on  Thales,  who  enunciated   the  first  proposition,  as 
well  as  many  others. 

5.  A  recreation  problem2  proving  that  every  triangle  is  isosceles, 
by  the  pupil  with  the  highest  grade. 

6.  Model  demonstration  of  some  proposition  in  Book  I,  by  some 
pupil,  who  will  give  it  orally  and  without  a  figure. 

7.  Paper  on  the  use  of  the  protractor  for  measuring  angles. 

PART  III.    AN  EXPOSITION 

8.  Talk  on  the  application  of  the  truths  learned  in  Book  I  in 
the  problems  of  life,  as  surveying,  architecture,  and  designing. 

1  Excellent  material  may  be  found  in  Sykes'    "Source  Book  of  Problems  in 
Geometry";   Allyn  and  Bacon. 
8  Wentworth  and  Smith,  "Plane  Geometry" ;  Ginn  and  Co. 


THIRD  SECTION 

ADVANCED  MATHEMATICS 


CHAPTER  FOUR 

SPECIAL  METHODS  OF  SUPERVISED  STUDY  IN  HIGHER 
MATHEMATICS 

Intermediate  and  Advanced  Algebra.  —  The  subjects  con- 
sidered in  this  chapter  are  usually  offered  to  pupils  in  the  last 
two  years  of  their  high  school  program.  These  pupils,  through 
the  study  of  elementary  algebra  and  plane  geometry,  which 
it  is  assumed  have  been  taught  on  the  supervised  study  plan, 
should  have  mastered  by  this  time  the  technic  of  how  to 
study,  and  should  be  able  to  handle  this  advanced  work  with- 
out the  detailed  directions  heretofore  necessary.  Consequently, 
the  inspirational  preview  and  how  to  study  lessons  will  only 
rarely  be  required.  The  general  plan  outlined  in  the  preced- 
ing lessons  may  be  followed  if  preferred,  but,  since  the  ideal 
method  would  be  for  the  pupils  to  advance  individually  and 
as  rapidly  as  they  are  capable  of  doing,  the  suggested 
lessons  herewith  given  will  be  based  on  this  plan.  With  this 
in  mind,  and  recognizing  that  pupils  electing  these  courses  will 
be  those  having  more  or  less  marked  mathematical  ability, 
we  shall  naturally  expect  that  some  will  be  able  to  cover  part 
if  not  all  the  subject  matter  of  intermediate  and  advanced 
algebra  during  the  semester. 

In  the  same  manner  and  with  the  same  expectation  we  shall 
treat  solid  geometry  and  plane  trigonometry  as  a  unit.  The 
time  spent  on  each  course  being  assumed  to  be  twenty  weeks, 
the  scheme  will  probably  work  out  about  as  follows :  If  at 
the  end  of  the  first  ten  weeks,  some  have  covered  at  least  three 

141 


142    Supervised  Study  in  Mathematics  and  Science 

fourths  of  the  work  in  intermediate  algebra  or  the  same  amount 
in  solid  geometry,  they  will  be  expected  to  complete  both 
courses  by  the  end  of  the  term.  Those  who  have  not  advanced 
so  far  will  finish  the  original  course  and  spend  the  remaining 
time  in  review.  This  plan  provides,  therefore,  for  each  pupil 
to  proceed  as  rapidly  as  he  is  capable  of  doing  and  yet  with 
no  detriment  to  himself  if  he  finds  he  is  unable  to  complete 
both  courses.  The  work  will  thus  resolve  itself  largely 
into  individual  instruction,  with  a  minimum  of  class  demon- 
stration. 

Method.  Since  a  large  part  of  intermediate  algebra  is  a 
review  of  elementary  algebra,  and  since  some  pupils  will  need 
but  little  review  and  drill  while  others  will  require  more,  after 
a  preliminary  explanation  of  the  plan  for  individual  advance- 
ment, a  test  or  series  of  tests  covering  the  work  of  the  elemen- 
tary course  should  be  given  to  enable  the  teacher  to  judge  just 
what  preliminary  or  review  study  is  necessary.  This  will 
naturally  vary  with  the  individual  pupils  and  therefore  resolve 
itself  at  once  into  personal  direction.  Those  showing  a  per- 
fect knowledge  of  elementary  algebra  will  immediately  be  set 
at  work  on  the  first  advanced  topic ;  those  displaying  a 
mastery  of  all  except  a  particular  topic  (for  example,  com- 
pleting the  square)  should  be  assigned  work  on  this  unit  of 
instruction,  and  so  on.  Some  may  have  forgotten  many  of 
the  details  of  the  preceding  course  and,  if  there  be  enough  of 
such  pupils,  they  may  easily  be  separated  from  the  others  and 
given  some  class  instruction.  In  other  words,  each  pupil 
should  have  his  knowledge  of  elementary  algebra  analyzed, 
and  he  should  be  set  to  work  upon  the  things  he  has  forgotten 
or  knows  imperfectly.  By  intensive  work  with  the  individ- 
ual pupils,  such  deficiencies  soon  may  be  overcome. 


Study  in  Higher  Mathematics 


FIGURE  V 

Keeping  a  Card  Index  of  the  Pupil's  Advancement.  The 
teacher  should  keep  a  record  of  each  pupil,  preferably  on  the 
card  index  plan,  noting  thereon  what  each  pupil  is  doing,  when 
he  has  mastered  each  unit,  and  other  pertinent  data.  If  the 
units  of  instruction  of  the  courses  in  algebra  have  been  eval- 


144    Supervised  Stttdy  in  Mathematics  and  Science 

uated  in  some  such  manner  as  was  done  prefatory  to  the 
illustrative  lessons  in  elementary  algebra  in  Section  I,  this 
record  may  be  kept  in  a  very  efficient  way.  The  six  column 
charging  card,  used  by  librarians,  has  been  found  very  satis- 
factory in  this  connection.  (See  Figure  V.)  In  the  first 
column  is  checked  the  number  of  the  unit  of  instruction,  in 
the  second  column  is  noted  the  date  it  is  begun,  and  in  the  next 
column  the  date  it  has  been  completed  to  the  satisfaction  of  the 
teacher.  The  next  three  columns  may  be  utilized  for  memo- 
randa concerning  the  difficulties  experienced,  test  marks,  etc. 
One  of  these  cards  for  each  pupil  is  easily  administered  and 
will  effectually  enable  the  teacher  to  "  keep  tab  "  on  each  pupil. 

When  the  class  assembles,  each  pupil  will  immediately 
begin  work  on  his  individual  task.  He  will  request  the  teacher 
to  render  assistance  if  it  is  needed.  The  teacher,  on  the  other 
hand,  when  not  so  employed,  will  pass  from  pupil  to  pupil, 
noting  progress,  checking  results,  and  outlining  advanced  work. 
The  schoolroom  thus  becomes  a  busy  workshop,  with  every- 
one engaged  on  his  particular  problems,  and  no  time  is  wasted. 
The  pupils  will  respond  eagerly  to  this  method,  because  they 
will  feel  that  they  must  proceed  just  as  rapidly  as  they  can 
with  thoroughness. 

The  mastery  of  each  topic  by  the  pupils  may  be  effectually 
checked  by  the  time-honored  test,  but  it  should  be  given  by 
units  and  as  soon  as  the  pupil  considers  himself  proficient 
enough  to  take  it.  The  efficiency  of  the  correspondence 
school  is  thus  realized  with  the  additional  benefits  derived  from 
personal  contact  with  the  teacher  and  the  momentum  and 
inspiration  of  the  classroom  and  its  environment.  Whatever 
additional  work  it  may  cause  the  teacher  is  counterbalanced 
by  the  high  rate  of  efficiency  and  the  elimination  of  ordinary 


Study  in  Higher  Mathematics  145 

routine  of  the  recitation.  This  will  offset  any  possible  hardship 
of  added  labor.  It  is  not  work  that  wears  out  the  teacher,  but 
worry;  if  the  results  are  satisfactory,  very  few  teachers  will 
be  found  to  complain  of  the  effort  demanded. 

Solid  Geometry  and  Plane  Trigonometry.  —  It  is  again 
assumed  that  pupils  electing  these  courses  have  had  those 
subjects  already  treated.  Through  the  process  of  the  sur- 
vival of  the  fittest,  the  class  will  undoubtedly  be  composed  of 
those  pupils  with  natural  mathematical  ability.  The  plan,  as 
already  outlined,  will  therefore  prevail  with  some  modifications, 
due  to  the  nature  of  these  subjects. 

With  all  superfluous  work  eliminated,  pupils  of  natural 
ability  ought  to  be  able  to  master  both  subjects  in  twenty 
weeks'  time.  The  ability  to  comprehend  and  master  the  text- 
book demonstrations  having  been  acquired  in  plane  geometry, 
the  study  of  these  in  solid  geometry  should  be  very  easy.  A 
few  class  explanations  of  the  use  of  the  third  dimension  might 
be  necessary  at  first,  until  the  pupils  realize  and  are  able  to 
visualize  the  figures.  The  method  of  proof  is  practically 
identical  with  that  of  the  demonstrations  in  plane  geometry 
and  the  number  employed  is  much  less ;  and  the  amount  of 
work  in  originals  is  greatly  reduced.  Since  not  all  pupils  are 
likely  to  complete  both  solid  geometry  and  trigonometry,  the 
work  in  the  latter  may  well  be  treated  individually  as  each 
pupil  reaches  it. 

Method.  As  soon  as  a  pupil  considers  himself  master  of  a 
demonstration,  he  may  call  the  teacher  to  his  side  and  either 
prove  the  proposition  orally  or  on  paper  as  the  teacher  elects. 
For  the  work  on  circles,  a  large  spherical  blackboard  should 
be  used  by  the  pupils  and  his  figures  thereon  should  be  ex- 
plained in  detail.  The  use  of  stereoscopic  views  for  showing 


146    Supervised  Study  in  Mathematics  and  Science 

the  more  complicated  figures  will  do  much  to  enable  the 
pupil  to  grasp  their  construction.  These  may  be  studied  by 
the  pupil  whenever  there  is  need  to  encourage  him  to  work 
them  out  for  himself.  When  called  on  for  aid,  the  teacher 
should  as  far  as  possible  confine  his  help  to  directing  questions, 
which  will  enable  the  pupil  to  discover  the  answer  to  his  own 
inquiry. 

Summary.  —  The  success  of  the  above  plan  for  the  advance- 
ment of  each  pupil  as  rapidly  as  he  is  able  to  master  the  sub- 
ject rests  on  two  essentials:  (i)  a  thorough  mastery  of 
the  technic  of  how  to  study,  developed  through  a  sys- 
tem of  supervised  study  from  the  early  grades,  and  (2)  a 
teacher,  thoroughly  acquainted  with  this  system  and  with  an 
intelligent  grasp  of  his  subject,  who  is  sympathetic  and  alert 
to  the  possibilities  of  the  effective  administration  of  this 
system.  With  such  qualifications  of  the  teacher  and  normal 
intelligence  and  application  on  the  part  of  pupil,  the  results 
will  be  highly  successful,  because  they  are  the  logical  and 
inevitable  culmination  of  the  system  of  supervised  study. 


PART   TWO 
SCIENCE 


CHAPTER  FIVE 

THE  MANAGEMENT  OF  THE  SUPERVISED  STUDY  PERIOD  IN 

SCIENCE 

Supervised  study  in  its  last  analysis  is  essentially  the  method 
of  the  laboratory,  and  therefore  science  should  and  does  lend 
itself  ideally  to  its  application.  The  study  of  a  textbook  is 
more  or  less  subordinated  to  the  study  of  actual  materials, 
individual  experimentation  supplanting  to  a  marked  degree 
the  assumption  of  certain  facts  and  conclusions  because 
stated  as  such  in  some  book.  In  other  words,  the  pupil  is 
taught  to  verify  what  the  author  has  said. 

The  study  of  science,  largely  perhaps  because  of  its  peculiar 
method  of  approach,  has  always  been  more  or  less  popular 
with  young  people.  Growing  boys  and  girls  are  enabled 
through  the  actual  handling  of  and  experimentation  with  real 
things  to  give  expression  to  their  very  active  desire  to  do  things, 
and  an  interest  is  aroused  which  can  never  be  imparted  through 
the  mere  reading  of  the  printed  page.  It  is  for  this  very  rea- 
son that  manual  training,  domestic  art,  and  allied  subjects  are 
especially  attractive.  The  adolescent  youth  yearns  to  deal 
with  "  objective  realities " *  and,  when  they  assume  in 
addition  some  practical  aspect,  their  study  seems  peculiarly 
valuable. 

1  Lloyd  and  Bigelow,  "The  Teaching  of  Biology";  Longmans,  Green  and 
Co.,  1914. 

149 


1 50    Supervised  Study  in  Mathematics  and  Science 

Laboratory  work,  however,  needs  to  be  carefully  supervised 
if  the  student  is  to  acquire  real  benefits  from  it,  as  aimless 
tinkering  with  apparatus  and  equipment  may  become  simply 
a  waste  of  time.  It  is  for  this  reason  that  the  supervised 
study  period  in  science  is  particularly  well  suited  to  show  the 
benefits  of  the  system.  Pupils  must  be  directed  in  their 
scientific  study  even  more  carefully  than  in  the  case  of  mere 
book  study  and  with  correspondingly  larger  results.  They 
must  be  taught  how  to  test  and  how  to  properly  draw  con- 
clusions; how  to  check  their  personal  observation  with  that 
of  others ;  how  to  use  various  kinds  of  supplementary  material 
and  how  to  judge  it ;  how  to  interpret  printed  directions  and 
to  form  unprejudiced  opinions ;  in  fine,  how  to  study. 

Much  that  has  been  written  in  Chapter  One  is  equally 
applicable  in  connection  with  the  work  in  science. 

The  Time  Schedule.  —  The  length  of  the  period  being 
assumed  to  be  sixty  minutes,  the  division  of  the  time  for  the 
regular  recitation  periods  may  be  made  as  follows : 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

When  laboratory  work  is  desired,  however,  the  entire  sixty 
minutes  may  well  be  devoted  to  the  experimental  work  and  its 
recording.  If  five  or  ten  minutes  be  taken  at  the  opening  of 
the  hour,  however,  to  explain  what  the  object  of  the  experiment 
is,  it  will  be  time  well  spent.  And,  again,  much  will  be  gained 
if  a  few  minutes  at  the  close  of  the  period  be  used  to  clinch  the 
essential  results  of  the  laboratory  work.  These  variations 
will  be  more  fully  explained  in  connection  with  the  illustrative 
lessons. 


Study  Period  in  Science  151 

The  Assignment  Sheet.  —  The  use  of  the  assignment  sheet, 
as  explained  in  the  foregoing  chapter,  is  strongly  recommended 
for  the  science  teacher  in  order  that  he  may  have  a  clear  out- 
line of  just  what  he  expects  to  accomplish  within  the  hour. 

The  threefold  assignments  are  also  urged  if  the  highest 
efficiency  of  the  class  is  to  be  developed.  While  there  is 
certain  work  that  must  of  necessity  be  demanded  of  all,  there 
should  always  be  some  arrangement  made  by  which  those  who 
are  particularly  gifted  may  be  led  to  do  advanced  work  and 
thus  proceed  at  a  maximum  rate  and  so  become  more  proficient 
in  their  knowledge  of  the  subject.  This  may  take  the  line  of 
extra  study,  supplementary  reading,  use  of  special  apparatus, 
making  charts,  classifying  museum  materials,  attending  to 
equipment,  making  of  simple  apparatus,  etc.  Obviously  the 
teacher  who  is  satisfied  with  a  minimum  or  average  amount 
of  work  being  done  alike  by  all  is  failing  to  realize  that  the 
schools  should  develop  each  pupil  according  to  his  individual 
ability.  This  possibility  of  leading  all  pupils  to  do  their  best, 
of  inspiring  them  to  strive  always  for  maximum  results,  is  in- 
deed one  of  the  strongest  arguments  in  favor  of  supervised 
study  in  science  as  well  as  in  all  departments.  There  is  no 
other  method  of  conducting  classes  which  so  nearly  approaches 
the  ideal  of  scientifically  educating  all  the  children  to  the  limit 
of  the  individual  capacity  of  each. 

Mimeographed  Sheets.  —  The  use  of  laboratory  manuals 
is  not  recommended  since  they  naturally  contain  many  things 
that  the  live  teacher  will  find  it  necessary  to  change,  on  account 
of  the  apparatus  at  hand  or  because  he  will  find  that  many 
things  he  wishes  to  incorporate  are  not  mentioned.  Mimeo- 
graphed sheets,  which  allow  the  individuality  of  the  teacher  to 
be  displayed,  are  much  to  be  preferred.  These  may  be 


152    Supervised  Study  in  Mathematics  and  Science 

changed  from  time  to  time  as  the  teacher  develops  new  ideas. 
In  lieu  of  these,  the  blackboard  may  be  used,  but  less  advan- 
tageously. 

Too  much  should  not  be  stated  on  these  sheets.  One  of 
the  gravest  faults  with  laboratory  manuals  is  their  complete- 
ness ;  too  often  they  leave  nothing  for  the  pupil  to  do  but  to 
"  press  the  button."  If  the  pupil  is  really  to  grow  from  day  to 
day  in  his  ability  to  perform  experiments  and  draw  correct 
conclusions,  much  must  be  left  to  his  own  ingenuity  and  re- 
sourcefulness. In  cases  where  the  pupil  seems  to  have  this 
ability  undeveloped,  the  necessary  explanations  may  better  be 
made  to  him  individually ;  thus  it  becomes  possible  for  each 
pupil  to  rely  to  a  large  extent  on  his  own  endeavors.  In  other 
words,  the  directions  should  be  mainly  suggestive,  and  details 
should  be  supplied  by  the  teacher  or  pupil  as  the  case 
requires. 

The  pupils  should  also  be  encouraged  to  construct  as  much  of 
their  apparatus  as  they  are  able  to  make.  Every  well-equipped 
laboratory  should  have  a  workbench,  fully  supplied  with 
ordinary  tools,  materials,  etc.,  so  that  the  pupil  may  have  an 
opportunity  to  develop  whatever  ingenuity  he  possesses. 

The  teacher  should  also  be  careful  not  to  do  too  much  him- 
self. His  ingenuity  in  constructing  apparatus  and  performing 
experiments  may  best  be  displayed  by  his  ability  to  develop 
ingenuity  and  performance  in  his  pupils.  The  pupil  should 
be  led  to  draw  the  proper  conclusions  and  not  told  what  con- 
clusions he  should  draw.  The  latter  is  rather  dogmatic  teach- 
ing but  the  former  may  well  be  called  superb  leadership. 


FOURTH  SECTION 
BIOLOGY 


CHAPTER  SIX 

DIVISIONS   OF  BIOLOGY 

In  evaluating  a  course  in  biology,  since,  from  the  very  nature 
of  the  vast  amount  of  available  material,  it  is  impossible  to 
cover  all  the  phases,  it  is  best  to  select  certain  topics  in  botany, 
zoology,  and  physiology  which  illustrate  the  fundamental 
principles  of  life.  This  method  at  the  same  time  will  give  the 
boys  and  girls  some  important  first-hand  knowledge  of  plant 
and  animal  life,  with  special  emphasis  on  the  economic  and 
hygienic  sides.  Since  authorities  will  naturally  differ  on  the 
topics  that  should  be  included,  the  divisions  as  here  made  may 
be  open  to  criticism,  but  for  the  sake  of  concreteness  those 
outlined  by  the  State  Department  of  Education  of  the  State  of 
New  York  are  followed,  with  a  few  modifications. 

UNITS  OF  INSTRUCTION 

I.   Introductory  topics. 

PLANT  BIOLOGY 

II.  Seeds  and  seedlings. 

III.  The  cellular  structure  of  living  plants. 

IV.  Roots. 
V.  Stems. 

VI.  Leaves. 

VII.   Flowers  and  fruits. 
VIII.   Forests  and  forest  products. 


156    Supervised  Study  in  Mathematics  and  Science 

ANIMAL  BIOLOGY 

IX.  Insects. 

X.  Crustaceans. 

XI.  Fishes. 

XII.  The  frog. 

XIII.  Birds. 

XIV.  Mammals. 

HUMAN  BIOLOGY 

XV.  Foods,  stimulants,  narcotics. 

XVI.  Bones  and  muscles. 

XVII.  Organs  of  digestion  and  their  functions. 

XVIII.  Blood  and  circulation. 

XIX.  Respiration. 

XX.  Excretion. 

XXI.  Bacteria  and  sanitation. 

XXII.  Additional    topics,   including   the   nervous   system, 
special  senses,  first  aid. 

Below  is  given  a  suggestive  time  table  for  the  year's  work  in 
biology.  This  will  of  course  vary  with  local  conditions  but 
is  given  merely  for  its  possible  suggestive  value : 

First  twelve  weeks Plant  biology 

Next  twelve  weeks Animal  biology 

Next  twelve  weeks Human  physiology 

Last  four  weeks Review 

LESSON  I 

THE  INSPIRATIONAL  PREVIEW 

Purpose.  —  The  purpose  of  such  a  lesson  is  to  arouse  the 
interests  of  the  child  in  the  study  of  biology.  There  is  no  sub- 
ject in  the  whole  program  of  studies  which  may  have  as  many 
practical  applications  to  the  welfare  of  the  pupil  as  does  the 


Divisions  of  Biology  157 

study  of  biology.  But  arousing  the  pupils  to  an  appreciation 
of  these  values  and  incidentally  to  its  many  pleasures  and 
interesting  phases  necessitates  that  the  teacher  make  some 
effort  to  predispose  the  child's  mind  to  receive  its  valuable 
lessons.  This  first  meeting  with  the  class  presents  this  op- 
portunity. 

Method.  Naturally,  the  method  of  conducting  an  inspira- 
tional lesson  will  vary  with  the  teacher,  since  it  will  reflect  his 
personal  ingenuity,  but  for  the  purpose  of  possible  suggestion, 
the  following  scheme,  which  has  been  successfully  used  by  the 
author,  is  presented  herewith. 

Previous  to  the  assembling  of  the  class  for  the  first  time, 
place  a  small  quantity  of  grass  and  some  grasshoppers  in  a 
fine-screened  cage.  As  soon  as  the  class  is  assembled,  call 
attention  to  the  cage  and  state  that  it  contains  illustrations, 
in  a  way,  of  the  purpose  of  the  study  of  biology.  Ex- 
plain that  biology  is  the  study  of  life,  animal  and  vegetable,  and 
its  practical  applications  to  the  existence  of  man.  Man  is  not 
only  dependent  on  these  two  kingdoms  of  nature  for  his  suste- 
nance, but  his  raiment,  his  health,  his  vocations,  and  his  avoca- 
tions are  largely  biologic  in  their  nature.  Draw  attention  to 
the  grasses  and  plants  in  the  cage  and  explain  that  the  vege- 
table kingdom,  of  which  they  are  examples,  furnishes  us  with 
food,  directly  or  indirectly,  clothing,  homes  in  which  to  live, 
and  determines  many  of  our  most  important  occupations ;  that 
some  plants  cause  sickness  and  others  offer  the  means  of  pre- 
venting or  curing  our  ills,  others  give  us  pleasures,  etc. 
Explain  that  in  order  to  know  just  how  these  things  come  about, 
it  behooves  us  to  know  how  the  plant  grows,  how  it  lives,  and 
how  it  reacts  upon  our  lives.  Tell  the  pupils  that  there  is 
within  this  cage  also  a  representative  of  the  animal  kingdom 


158    Supervised  Study  in  Mathematics  and  Science 

—  a  large  group  of  living  things,  on  which  in  a  large  way  we 
are  likewise  dependent  for  food,  raiment,  transportation, 
good  health,  and  also  many  of  the  pleasures  of  life.  Our  work 
in  biology  will  lead  us  into  a  systematic  study  of  animal  life 
and  its  part  in  the  scheme  of  living. 

Now  have  someone  come  forward  to  open  the  cage  and 
find  a  grasshopper  within.  The  difficulty  of  finding  him  in 
his  camouflaged  natural  habitat  will  give  the  teacher  an 
opportunity  to  speak  of  the  insect's  environment,  his  protective 
coloration  or  adaptation,  his  dependence  upon  the  plant  king- 
dom, and,  therefore,  the  close  connection  between  plants  and 
animals. 

Place  the  grasshopper  upon  the  floor  and  let  him  jump. 
Measure  the  length  of  the  jump  and  compare  it  with  his  size. 
Ask  some  pupil  how  high  he  can  jump  and  make  a  like  com- 
parison with  his  height.  The  wonderful  adaptability  of  the 
hind  legs  of  the  grasshopper  for  this  activity  will  appeal  to  the 
pupils  as  extraordinary,  as  it  indeed  is.  Mention  the  fact 
that  all  the  various  classes  of  animals  we  shall  study  have  like 
wonderful  adaptations  for  their  modes  of  life  and  also  that  they 
are  beautifully  made  for  the  part  they  are  to  play  in  the  scheme 
of  nature. 

Now  after  a  few  words  briefly  explaining  the  various 
functions  of  living  matter,  write  upon  the  board  the  following 
captions  which  may  be  said  to  form  the  background  of  our 
work: 

Classification. 

Habitat. 

Structure. 

Life  history. 

Adaptations. 

Functions. 


Divisions  of  Biology  159 

If  now  the  teacher  will  read  some  interesting  incident 
showing  the  wonderful  intelligence  of  some  insect,  as  the  ant 
from  Romanes'  "  Animal  Intelligence," l  or  the  fly  from 
Fabre's  "  The  Life  of  the  Fly," 2  or  the  wasp  from  Morey's 
"  Wasps  and  Their  Ways,"  ''•  he  will  arouse  an  intense 
desire  to  know  more  about  these  wonderful  little  creatures, 
and  the  year's  work  in  biology  will  begin  auspiciously  and  in- 
terest can  be  easily  maintained. 

Then  briefly  tell  the  story  of  the  Panama  Canal  and  its 
gigantic  failure  until  General  Goethals  attacked  the  biologic 
problems  and  made  the  region  safe  for  man  to  inhabit,  chang- 
ing failure  to  success.  Tell  about  the  work  of  Dr.  Koch  and 
his  discovery  of  the  cause  of  tuberculosis,  which  has  resulted 
in  its  subsequent  treatment ;  the  wonderful  work  of  Luther 
Burbank  in  producing  new  varieties  of  plants ;  the  investiga- 
tions of  Dr.  Wiley  in  regard  to  pure  food ;  the  work  of  the 
Federal  Bureau  of  Entomology  and  its  aid  to  the  farmer. 
Explain  that  in  every  case  it  was  the  application  of  biologic 
facts  that  has  made  life  more  enjoyable,  more  healthful,  and 
more  successful. 

With  such  a  bird's-eye  view  of  the  contributions  of  biology, 
the  pupils  will  not  fail  to  realize  that  here  is  a  subject  full  of 
the  practical  and  the  interesting ;  and  they  will  actually  feel 
the  desire  to  master  this  subject  in  so  far  as  they  are 
able. 

If  time  permits,  a  rapid  survey  of  the  course  may  be  made. 
Explain  that  first  we  shall  make  a  study  of  the  elements  that 
enter  into  all  life,  both  animal  and  plant ;  then  we  shall  study 
plants  from  the  seedling  to  the  matured  plant ;  animals  from 
the  simple  one-celled  species,  which  can  only  be  seen  under  the 
1  D.  Appleton  and  Co.  2  Dodd,  Mead  and  Co. 


160    Supervised  Study  in  Mathematics  and  Science 

microscope,  up  through  the  crustaceans,  the  insects,  the  fishes, 
birds,  and  mammals;  and  finally  we  shall  make  a  study  of 
man,  the  most  wonderful  and  highest  creation  of  all. 

The  pupils  should  be  informed  that  they  will  be  expected  to 
bring  specimens  to  class,  from  time  to  time,  to  add  to  the 
school's  collection  of  natural  history  materials.  Tell  them 
to  be  on  the  lookout  for  articles  relating  to  biology,  such  as  are 
appearing  constantly  in  newspapers,  magazines,  and  books. 
Tell  them  that  in  addition  to  the  study  of  their  textbook  we 
shall  do  more  or  less  experimentation  and  investigation  of 
actual  specimens  in  the  laboratory  and  in  their  natural  envi- 
ronment. 

The  teacher  must  be  careful  not  to  make  this  preview  too 
technical  or  too  stilted.  It  is  very  important  to  make  the 
pupils  forget  that  they  are  about  to  become  formal  students 
of  a  new  subject  which  may  contain  a  good  deal  of  dry,  uninter- 
esting drill  on  facts.  Young  people  at  this  age  are  naturally 
more  or  less  inquisitive  and  something  which  will  make  keener 
this  instinct  for  the  novel  or  the  unknown  will  be  worth  while 
if  at  the  same  time  it  arouses  within  them  a  sincere  desire  to 
learn. 

The  above  suggested  plan  may  be  objected  to  on  the  ground 
that  it  smacks  too  much  of  the  idea  of  nature  study  and  is  too 
childish.  Are  we  not  dealing  with  young  children  of  immature 
minds  and  is  not  biology  in  fact  nature  study  with  the  stress 
on  its  applications  to  life?  In  the  best  sense  biology  is  or 
should  be  the  study  of  natural  history  with  the  underlying 
thought  of  the  continuity  of  life  processes  and  functions,  and 
its  applications  to  the  whole  scheme  of  nature.  Both  experi- 
ence and  pedagogical  considerations  seem  to  justify  this  view- 
point. 


Divisions  of  Biology  161 

LESSON  II 

UNIT   OF  INSTRUCTION  I.  —  INTRODUCTORY  TOPICS 

LESSON  TYPE.  —  A  LESSON  ON  How  TO  STUDY 
Program  or  Time  Schedule 

The  Assignment 10  minutes 

The  Study  of  the  Assignment 50  minutes 

Purpose.  —  The  purpose  of  this  lesson  is  to  introduce  the 
pupil  to  the  study  of  biology  through  some  preliminary  experi- 
ments with  chemical  elements  and  compounds.  The  textbook 
is  at  once  to  be  supplemented  with  the  actual  handling  and 
examination  of  biological  materials,  thus  illustrating  the 
laboratory  method  to  be  followed  more  or  less  throughout  the 
course.  It  is  desirable  to  make  clear  at  the  outset  that  the 
textbook  is  in  the  nature  of  a  guide,  to  explain  and  organize  our 
study.  Just  as  guidebooks  in  travel  may  not  only  serve  as  a 
prospectus  of  what  we  may  expect  to  find  and  see,  but  also  as 
a  means  of  interpreting  those  things  which  we  see  and  wish  to 
know  more  about,  so  our  textbook  in  biology  will  guide  us  in 
our  work  to  discover  certain  things  and  will  supplement  what 
we  see  with  data  concerning  which  we  would  otherwise  remain 
in  ignorance. 

The  Review.  —  This  will  usually  take  the  form  of  clinching 
firmly  in  the  mind  of  the  child  the  lessons  learned  previously, 
and  the  process  should  be  pointed,  short,  and  clarifying.  As 
this  is  the  first  lesson  besides  the  inspirational  preview,  the  re- 
view may  be  dispensed  with  to-day. 

The  Assignment.  —  In  all  laboratory  work  the  assignment 
may  well  take  but  a  few  minutes  and  will  be  used  to  explain 
the  nature  of  the  new  work.  In  the  present  case,  it  will  suffice 


162    Supervised  Study  in  Mathematics  and  Science 

to  explain  that  we  are  going  to  spend  the  period  examining 
some  material  as  to  its  physical  characteristics  and  as  to  what 
it  will  do  under  certain  manipulations. 

Notebooks.  Explain  that  the  use  of  notebooks,  or  some 
means  of  recording  our  investigations,  is  necessary  in  all 
laboratory  work  in  order  that  we  may  set  down  scientifically 
the  facts  we  learn  and  that  we  may  have  them  for  future 
verification  and  review.  Data  so  recorded  must  be  accurate 
and  as  clear  as  we  know  how  to  make  them.  Explain  that 
some  system  should  be  followed  which  will  help  to  make 
the  data  clear.  Each  of  the  following  heads  should  be 
employed : 

1.  Date. 

2.  Object. 

3.  Material. 

4.  Manipulations. 

5.  Results. 

All  pupils  should  be  supplied  with  some  form  of  notebook, 
the  kind  necessarily  varying  with  the  individual  taste  of  the 
teacher.  The  end-open  loose-leaf  notebook,  size  about  4X6 
inches,  is  recommended.  This  makes  a  compact  little  book 
which  may  be  carried  easily  in  the  pocket  or  bag  on  field 
excursions.  State  that  these  books  will  be  collected  from 
time  to  time  and  examined  by  the  teacher,  and  emphasize  the 
importance  of  keeping  them  as  neat  as  possible  and  of 
always  having  them  ready  for  use. 

The  Study  of  the  Assignment.  —  There  should  be  placed 
upon  the  desk  or  table  of  each  pupil  all  the  materials  which  are 
to  be  studied.  A  little  careful  planning  will  obviate  wasting 
of  time  during  the  period.  In  the  present  instance,  each  pupil 


Divisions  of  Biology  163 

should  be  supplied  with  small  pieces  of  carbon  (charcoal),  iron, 
sulphur,  and  phosphorus  (in  water). 

Procedure.  Tell  the  pupils  to  open  their  texts  and  read  the 
paragraph  on  elements  and  compounds.  As  soon  as  they  have 
had  time  to  read  it  through  understandingly,  have  them  close 
their  books  and  question  them  on  what  they  have  read.  The 
teacher  may  well  at  this  point  expand  somewhat  the  state- 
ments of  the  author,  being  careful  not  to  confuse  chemical  with 
physical  compounds. 

Then  tell  the  pupils  to  examine  the  piece  of  charcoal  with- 
out referring  to  the  book,  and  after  a  few  minutes  question 
them  as  to  its  characteristics.  Try  to  get  them  to  tell  what 
they  have  found  out  about  the  charcoal  without  your  asking 
too  many  questions.  Different  pupils  may  be  called  upon  to 
add  something  new  to  the  peculiarities  or  characteristics  al- 
ready given,  until  all  the  important  ones  have  been  mentioned. 
If  this  method  fails  to  elicit  all  the  desired  information,  it  may 
become  necessary  to  ask  definite  questions,  such  as,  What  is  its 
taste? 

Repeat  the  process  with  the  piece  of  iron,  and  then  have  the 
pupils  compare  the  two  elements  for  similarities  and  differ- 
ences. Insist  on  the  results  being  stated  in  sentence  form 
with  due  regard  to  the  English  used.  Ask  them  to  tell  in  what 
forms  each  of  the  two  may  be  found  and  draw  out  therefrom, 
if  possible,  the  significance  of  these  characteristics.  For 
instance,  they  have  discovered  that  charcoal  is  soft  and  leaves 
a  mark  when  drawn  across  paper.  This  quality  has  given 
rise  to  charcoal  pencils  used  by  artists.  A  similar  character- 
istic of  graphite  has  been  utilized  in  the  making  of  so-called 
lead  pencils.  Iron  is  found  to  be  tough  ;  how  has  this  char- 
acteristic been  utilized  in  making  iron  stoves,  rails,  etc.  ? 


164    Supervised  Study  in  Mathematics  and  Science 

After  they  have  done  this,  let  them  open  their  books  and 
read  over  what  the  author  has  to  say  concerning  these  two 
elements.  Someone  might  take  another  text  in  biology  and 
read  aloud  to  the  class  what  the  author  has  to  say  concerning 
this  subject.  Step  to  the  board  and  write  a  list  of  things  made 
from  each  element,  items  being  suggested  by  the  pupils  them- 
selves. It  is  extremely  important  to  have  the  pupils  feel  that 
they  are  furnishing  the  data  and  that  the  teacher  is  merely 
leading  the  way  and  correcting  any  wrong  deductions. 

Recording  the  Experiments.  The  next  step  is  to  record  in 
the  notebooks  the  results  of  this  exercise.  Have  the  pupils 
open  their  notebooks  and,  following  the  order  noted  above, 
make  their  records.  The  one  on  charcoal  should  appear 
something  like  this : 

PLAN  OF  NOTEBOOK 
Date:  September  4,  192-. 
Object:  To  find  the  characteristics  of  carbon. 

Materials :  Piece  of  charcoal,  a  dish  of  water,  a  match,  a  knife, 
paper. 

Experiment :  I  took  the  charcoal  in  my  hand,  noted  its  weight, 
color,  feel,  taste,  odor,  and  texture.  I  put  it  in  some  water  to  see 
whether  it  would  dissolve,  I  tried  to  make  it  burn,  I  tried  to  cut 
it  with  a  knife,  and  I  rubbed  it  upon  some  paper. 

Results :  I  found  that  it  was  light  in  weight,  black  in  color,  felt 
rough,  had  no  taste  or  odor,  and  looked  to  be  rather  porous.  It 
will  not  dissolve  in  water,  burns  with  a  glow,  and  makes  a  mark  on 
paper. 

Some  other  forms  of  carbon  are :  graphite  (used  in  lead  pencils) ; 
diamonds. 

Carbon  results  from  burning  wood,  as  the  match,  and  is  some- 
times found  free  in  nature. 


Divisions  of  Biology  165 

Next,  the  pupil  should  take  up  the  study  of  sulphur  in  a 
similar  way.  The  work  on  phosphorus  will  need  to  be  guided 
more  definitely  by  the  teacher,  due  to  its  peculiar  character- 
istics. After  the  four  elements  have  thus  been  studied  and 
written  up  in  the  notebooks,  if  time  permits,  the  pupils  should 
study  the  paragraph  on  oxygen  and  the  air,  after  which  the 
same  order  of  study  may  be  followed. 

It  is  very  important  to  make  haste  slowly  in  this  new  work. 
The  pupils  are  immature  and  cannot  grasp  too  many  new 
principles  at  one  time.  Each  phase  should  be  covered  thor- 
oughly, with  all  the  variations  that  it  is  possible  to  make.  The 
actual  material  in  the  textbook  should  be  supplemented  by 
other  books,  by  the  pupils'  experience  and  knowledge  and,  if 
necessary,  by  the  teacher.  The  illustrations  and  applications 
should  be  as  varied  and  comprehensive  as  possible,  for  this  is 
the  way  by  which  the  pupils  will  learn  to  study. 

LESSON  III 
UNIT  OF  INSTRUCTION   I.  —  INTRODUCTORY  TOPICS 

LESSON  TYPE.  —  AN  INDUCTIVE  LESSON 

Program  or  Time  Schedule 

The  Review 20  minutes 

The  Assignment 15  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  The  work  to  date  has  been  on  the  prelimi- 
nary experiments.  Since  we  are  now  to  start  on  a  new  topic  or 
unit  of  recitation,  it  may  be  best  to  spend  a  little  longer  time 
than  is  given  usually  to  review,  in  clinching  the  lessons  already 
studied.  This  may  take  the  manner  of  a  short,  snappy  quiz. 


1 66    Supervised  Study  in  Mathematics  and  Science 

In  addition,  we  might  make  a  formal  testing  of  some  phase  of 
the  work  previously  studied.  These  twenty  minute  reviews 
occurring  every  four  or  five  days,  partly  oral  and  partly  written, 
are  much  better  than  formal  written  examinations  of  length 
and  given  at  longer  intervals.  These  reviews  test  the  knowl- 
edge of  the  class  on  many  of  the  details ;  and  through  the 
short  written  test,  they  cover  with  definiteness  some  particular 
feature  of  the  work  and  make  it  possible  for  the  teacher  to  know 
just  how  completely  the  pupils  have  grasped  important  points. 
The  question  or  questions  —  one  is  usually  sufficient  —  should 
be  broad  and  should  be  based  upon  some  definite  or  underlying 
principle.  For  example,  the  class  to-day,  after  the  oral  quiz, 
might  be  asked  to  write  a  paragraph  on  the  importance  of 
reducing  all  compounds  to  their  elements. 

Allow  ten  minutes  for  this  paragraph  and  then  collect  the 
papers.  These  should  be  carefully  examined  and  errors  noted, 
and  two  or  three  of  the  best  answers  read  aloud  to  the  class 
the  next  day.  If  none  is  satisfactory,  the  reasons  for  this 
fact  should  be  summarized  and  possibly  a  model  answer  writ- 
ten upon  the  board.  Pupils  will  thus  learn  how  to  give  sat- 
isfactory written  answers.  Since  written  examinations  as  a 
means  of  judging  proficiency  in  a  subject  are  bound  to  be  with 
us  for  some  time  to  come,  this  is  an  effective  method  of  pre- 
paring the  pupils  for  them.  By  practice  and  through  criticism 
each  pupil  will  learn  the  kind  of  complete  and  definite  state- 
ments that  will  be  expected  from  him  in  these  written  tests. 
As  already  noted,  fact  questions  need  not  be  emphasized  in 
written  tests  since  the  oral  quiz  is  just  as  satisfactory  and  is 
a  time  saver. 

The  Assignment.  —  Explain  that  many  of  the  lessons  in 
biology  will  be  in  the  nature  of  problems.  To-day's  problem 


Divisions  of  Biology  167 

is:  Are  two  plants  or  parts  ever  alike?  Our  study  will  be 
based  upon  illustrative  material  and  the  textbook. 

Tell  the  pupils  to  make  a  list  of  articles,  found  in  nature, 
which  are  similar  in  form.  After  a  few  minutes  ask  someone 
to  read  the  name  of  the  first  article  on  his  list.  It  might  be 
"  man."  Ask  whether  two  men  are  ever  exactly  alike.  The 
pupils  will  readily  answer  that  they  are  not.  Repeat  with 
others  on  the  lists,  as  dog,  potato,  stone,  apple,  etc.  The 
pupils  will  agree  that  all  seem  to  differ  in  some  particular  from 
each  other.  Ask  whether  things  in  the  artificial  world  so 
differ  from  one  another.  They  are  more  likely  to  be  seemingly 
alike  but,  even  in  the  case  of  the  textbooks  in  use  in  class,  each 
book  may  differ  somewhat  from  its  neighbor.  Some  may  have 
pages  uncut,  some  may  have  defects  in  binding,  some  may 
have  pages  poorly  printed,  etc. ;  yet  they  differ,  if  at  all,  in 
degrees  of  perfection.  Then  direct  the  pupils  to  study  the 
maple  leaves  which  will  have  been  placed  in  advance  upon 
their  desks. 

The  Study  of  the  Assignment.  —  Each  pupil  having  been 
supplied  with  a  number  of  leaves,  ask  them  to  find  two  alike 
if  they  can.  They  may  exchange  with  each  other  if  they 
wish.  If  some  pupil  thinks  he  has  found  two  exactly  alike,  it 
may  be  necessary  for  the  teacher  to  point  out  some  difference 
in  veining,  hairiness,  markings,  color,  etc.  It  would  be  an 
excellent  thing  if  a  quantity  of  leaves  of  a  different  kind  could 
also  be  passed  around  for  examination.  The  greater  the 
variety  of  specimens  for  examination  the  more  scientific  and 
impressive  will  be  the  lesson  learned.  A  quantity  of  twigs 
from  the  same  tree  should  also  be  secured  for  a  like  investiga- 
tion. After  ample  time  has  been  allowed  for  all  to  satisfy 
themselves  that  no  two  specimens  are  entirely  alike,  have  the 


1 68    Supervised  Study  in  Mathematics  and  Science 

pupils  open  their  textbooks  and  read  what  the  author  says 
concerning  this  problem  (Bailey  and  Coleman's  First  Course 
in  Biology,  pp.  1-3). l  Draw  the  attention  of  the  class  to 
the  cut  which  also  justifies  the  same  conclusion,  noting,  how- 
ever, that  a  study  of  the  specimen  itself  is  to  be  preferred 
always  to  a  picture. 

Tell  them  to  read  carefully  the  last  sentence  and  then  ex- 
plain in  simple  manner  what  the  author  means.  This  word 
variation,  then,  is  the  key  word  of  our  lesson  to-day. 

The  pupil  may  never  have  appreciated  so  fully  before  the 
wonderful  ways  of  nature  as  he  will  now  when  he  comes  to  the 
realization  that,  of  all  the  billions  of  maple  leaves  in  the  world 
to-day,  as  well  as  in  all  the  years  gone  by,  there  are  never  two 
exactly  alike.  Here  is  an  opportunity  for  the  teacher  to  im- 
press the  child  with  an  appreciation  of  nature  through  scientific 
observation. 

The  observations  listed  on  page  3  of  the  text  with  regard  to 
size  may  well  be  taken  up  next  and  each  pupil  told  to  answer 
these  questions  concerning  one  of  the  leaves  he  has  before 
him.  If  time  permits,  it  would  also  be  beneficial  for  him  to 
note  these  facts  in  his  notebook. 

Before  this  is  done,  however,  the  result  of  the  day's 
investigations  should  be  noted  and  cast  into  a  form  of  state- 
ment ;  and  it  may  with  good  effect  be  written  upon  the  board 
with  colored  crayon.  Ask  someone  to  tell  what  conclusion  he 
has  drawn  from  the  day's  lesson.  If  the  pupil  has  been  alive  to 
the  problem  under  discussion  and  examination,  he  will  say, 
"  No  two  plants  or  parts  are  ever  alike." 

(N.B.    Note  the  title  of  the  chapter  referred  to.) 

1  The  Macmillan  Company. 


Divisions  of  Biology  169 

LESSON  IV 

UNIT   OF  INSTRUCTION   I.  —  INTRODUCTORY  TOPICS 

LESSON  TYPE.  —  AN  INDUCTIVE  LESSON 
Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  On  each  of  five  panels  of  blackboard,  have 
these  directions : 

Write  a  list  of  points  in  which  leaves  may  vary  from  each  other. 

Write  a  list  of  various  kinds  of  leaves. 

Make  a  list  of  flowers  which  do  not  at  all  resemble  each  other. 

How  do  people  resemble  each  other  ? 

How  do  people  differ  from  each  other? 

Assign  a  pupil  to  each  of  these  sets  of  instructions,  directing 
him  to  write  the  answers  upon  the  board.  Meanwhile,  ask 
someone  for  the  conclusion  reached  in  the  preceding  day's 
lesson,  and  ask  different  pupils  for  proof  of  its  accuracy. 

The  lists  having  now  been  completed,  read  each  exercise  and 
the  answers  given.  Remarks  may  be  necessary  to  amplify  or 
correct  mistakes ;  have  other  pupils  supply  this  material  where 
possible. 

The  Assignment.  —  Explain  that  the  new  lesson  is  to  be 
based  upon  a  study  of  a  continual  warfare  that  is  being  waged 
in  the  plant  and  animal  world ;  namely,  that  of  a  struggle 
for  life.  Remark,  while  one  may  go  to  the  mountains  for  the 
summer  and  meditate  upon  the  tranquillity  of  nature,  that  in 
fact  a  very  serious  strife  is  being  waged  by  every  tree  and 
flower  and  living  thing.  Each  is  making  a  desperate  struggle 


170    Supervised  Study  in  Mathematics  and  Science 

for  a  chance  to  live,  and,  just  as  each  is  successful  or  un- 
successful, will  the  existence  of  a  particular  flower  or  bird  be 
determined. 

Ask  the  pupils  whether  they  can  relate  any  experience 
in  observed  animal  or  plant  life  which  would  go  to  prove 
that  this  is  so.  Many  examples  in  the  animal  world  will 
come  to  their  minds  of  one  animal's  living  upon  another. 
After  a  few  illustrations  have  been  given,  ask  whether 
anyone  can  give  instances  from  the  vegetable  world.  This 
may  be  a  poser,  but  if  you  suggest  weeds,  the  class  will  follow 
the  clue. 

After  this  factor  in  plant  life  has  been  quite  fully  covered, 
explain  that  these  plants  and  animals  solve  their  problems  by 
various  means,  such  as  the  bird's  escaping  from  the  cat  by 
rapid  flight.  This  instance  will  set  the  pupils  thinking,  and 
the  teacher  will  need  the  next  few  minutes  to  give  them  an 
opportunity  to  state  similar  cases  of  adaptation  to  con- 
ditions. 

State  finally  that  these  conditions,  to  which  a  plant  or  animal 
is  obliged  to  adapt  its  habits  of  life  or  die,  is  called  its  environ- 
ment, and  bring  out  the  various  conditions  of  environment, 
such  as  climate,  food,  habitat,  etc. 

The  Study  of  the  Assignment.  —  The  advance  lesson  will  be 
the  chapter  on  "  The  Struggle  for  Life."  The  essential  fea- 
tures of  this  chapter  have  been  already  covered  in  our  study, 
but  without  any  reference  to  the  book  itself.  Now  through 
the  study  of  the  assignment  or  the  textbook,  the  pupils  are  to 
organize  this  material  into  a  clear,  concrete  increment  of 
knowledge.  Some  words  may  need  explanation;  pupils 
should  early  feel  that  the  dictionary  is  to  be  a  constant  help 
and  should  be  referred  to  frequently.  If  any  pupil  finds  some- 


Divisions  of  Biology  171 

thing  in  the  lesson  which  he  cannot  understand,  he  should 
raise  his  hand  and  call  the  teacher  to  his  aid. 

An  added  exercise  should  be  the  requirement  to  write  out  a 
list  of  at  least  ten  cases  of  struggle  for  existence,  ten  adapta- 
tions to  conditions,  and  ten  varieties  of  environment. 

LESSON  V 

UNIT   OF  INSTRUCTION   H.  —  SEEDS  AND   SEEDLINGS 

LESSON  TYPE.  —  A  How  TO  STUDY  LESSON 

Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 15  minutes 

The  Study  of  the  Assignment 35  minutes 

The  Review.  —  Each  pupil  having  been  assigned  some  plant 
for  special  report  to-day  on  the  essential  points  of  classification, 
the  review  may  be  spent  on  this  work.  Each  pupil  in  turn 
may  be  called  upon  to  name  the  plant  assigned  to  him  for  study, 
and  to  classify  as  annual,  pseud-annual,  plur-annual,  biennial, 
etc.,  and  to  state  clearly  his  reasons  for  his  answer.  (A  sugges- 
tive list  is  given  in  Bailey  and  Coleman's  First  Course  in  Biol- 
ogy, page  ip.)1 

The  Assignment.  —  Preparatory  Work  of  the  Teacher. 
Each  pupil  should  be  supplied  with  a  specimen  of  the  bean  and 
a  kernel  of  corn,  which  have  been  previously  soaked  in  water. 
Each  pupil  should  also  have  a  bottle  containing  a  solution  of 
iodine.  A  large  chart  showing  the  essential  parts  of  these  seeds 
should  also  be  in  full  view  of  the  class.  In  lieu  of  a  chart,  the 
teacher  might  make  large  drawings  upon  the  blackboard. 

1  The  Macmillan  Company. 


172    Supervised  Study  in  Mathematics  and  Science 

The  teacher  should  explain  in  the  beginning  that  the  lesson 
to-day  concerns  seeds,  their  essential  parts  and  their  functions. 
He  should  then  explain  how  to  open  a  seedling,  point  out  the 
embryo,  and  indicate  how  the  test  for  starch  should  be  made. 
He  should  also  explain  that  these  seeds  have  been  soaked  in 
water  in  order  to  study  them  more  easily  and  to  show  the 
effect  water  has  on  them. 

The  class  should  then  be  instructed  to  open  the  books 
and  to  study  the  text  carefully,  verifying  each  statement  from 
examination  of  the  specimens.  When  they  have  opened  their 
books,  but  before  they  start  their  work,  it  will  be  well  to  ex- 
plain the  correct  pronunciation  of  the  names  of  the  various 
parts  of  the  seed.  These  are  new  words  and  it  is  best  to  make 
sure  that  the  pupils  learn  from  the  first  just  how  they  are  to 
be  pronounced.  The  derivation  of  some  of  the  words  might 
well  be  given,  as  plumule  from  the  French  word  plume  mean- 
ing to  ascend  and  hence  given  to  that  part  of  the  seed  which 
will  rise  or  ascend.  Explain  that  monocotyledon  is  formed 
from  two  Greek  words,  mono,  meaning  one,  and  cotyledon, 
meaning  a  cup-shaped  hollow,  and  signifies  a  seed  having  one 
cotyledon.  Note  that  monoplane  is  an  airship  having  one 
wing  or  plane.  Again,  endosperm  is  formed  from  the  Greek 
adverb  meaning  around,  and  sperm,  which  signifies  the  life- giving 
element,  hence  the  word  denotes  that  which  is  around  or  encloses 
the  embryo  or  sperm.  The  pupils  thus  having  learned  the 
reasons  for  the  new  names  will  find  them  not  simply  unfamiliar 
and  unmeaning  terms  but  living  words,  and  they  will  experience 
no  particular  difficulty  in  learning  to  use  them  correctly. 

All  of  the  words  which  are  likely  to  give  trouble  should  be 
marked  in  the  teacher's  book  and  definitions  or  explanations 
of  them  prepared  prior  to  the  meeting  of  the  class.  It  is  this 


Divisions  of  Biology  173 

careful  preparedness  on  the  part  of  the  teacher  that  will  do 
more  than  anything  else  to  dispel  confusion  and  that  will 
result  in  the  interested  and  undivided  attention  of  the  as- 
sembled class. 

The  Study  of  the  Assignment.  —  The  pupils  will  now  begin 
the  study  of  the  new  work.  The  first  sentence  presupposes 
the  opening  of  the  seed  and  the  discovery  of  the  embryo. 
This  then  should  be  done,  the  bean  being  used  first.  The 
next  sentence,  we  shall  assume,  describes  the  three  essential 
parts  of  this  embryo.  With  the  direction  of  the  teacher  and 
the  aid  of  the  large  charts,  these  should  be  found  in  the  speci- 
men and  the  proper  name  for  each  learned.  If  any  pupil 
has  difficulty  in  finding  these  parts  in  his  specimen,  he  should 
call  the  teacher  at  once  to  his  side.  Let  it  be  definitely  under- 
stood that  no  one  is  to  proceed  until  he  has  covered  thoroughly 
each  individual  point  of  the  text. 

After  this  has  been  done  with  the  bean  seed,  tell  the  pupils 
to  open  up  the  kernel  of  corn  and  trace  its  embryo  through  a 
like  investigation.  The  teacher  should  follow  the  actual  work 
of  the  pupils  throughout,  keeping  pace  with  them  in  their 
study  and  supplying  any  details  necessary  to  make  the  work 
clear.  Ninety  per  cent  of  the  textual  work  in  a  science  must 
be  studied  in  this  painstaking  and  critical  manner,  nothing 
being  passed  by  until  mastered  and,  if  possible,  verified  from  a 
study  of  the  specimen  or  material  itself.  We  simply  cannot 
read  scientific  literature  over  rapidly,  looking  for  the  high 
spots,  as  is  sometimes  done  in  some  subjects,  but  each  state- 
ment must  be  closely  and  carefully  studied.  It  is  for  this 
reason  that  supervised  study  in  science  becomes  very  im- 
portant, and  why  study  without  it  often  results  in  superficial 
knowledge  or  utter  failure. 


1 74    Supervised  Study  in  Mathematics  and  Science 

Not  too  much  should  be  assigned  for  a  lesson.  Better  a 
little  well  studied  and  principles  well  grounded  than  to  attempt 
too  much.  An  excellent  method  of  varying  the  order 
suggested  in  the  above  time  schedule  would  be  to  use  the  last 
five  or  ten  minutes  of  the  period  for  a  review  of  the  essential 
points  studied  in  the  new  lesson,  thus  firmly  clinching  them. 
Step  to  the  chart,  for  instance,  and  call  on  someone  to  name 
the  different  parts  of  the  seed,  someone  else  to  tell  the  function 
of  each,  and  someone  else  to  spell  the  various  names. 

Before  the  class  is  excused,  some  of  the  unopened  soaked 
seeds  should  be  planted  in  a  suitable  box  and  placed  in  a  sunny 
place ;  some  similar  seeds  which  have  not  been  soaked  should 
also  be  planted  in  another  box  and  the  two  boxes  labeled,  one 
as  A  and  the  other  as  B.  It  is  suggested  that  some 
pupils  be  assigned  to  this  duty,  preferably  someone  who  has 
been  a  little  more  industrious  than  the  others  and  who  may 
have  completed  the  assignment.  This  will  serve  to  keep  all 
busy  and  will  give  some  recognition  to  the  more  rapid  workers. 
Speed  should  never  be  tolerated  in  place  of  thoroughness, 
however. 

If  there  be  time  for  any  more  work  during  the  period,  the 
pupils  may  be  instructed  to  make  a  drawing  of  the  opened 
seed  in  their  notebooks,  labeling  each  part.  Under  the  title  of 
Seed  Germination,  the  pupils  should  also  commence  a  note- 
book record  of  the  planting  of  the  soaked  and  unsoaked  seeds. 
The  record  will  be  added  to  later.  If  the  work  has  been  care- 
fully planned  in  advance,  all  of  the  required  details  can  be 
completed  within  the  hour ;  the  exact  amount,  however,  will 
necessarily  vary  with  teachers  and  classes.  Sufficient  work 
should  be  planned  by  the  teacher  for  an  emergency,  always 
keeping  the  time  limit  of  the  period  in  mind  so  that  there  may 


Divisions  of  Biology  175 

be  no  undue  hurry  and  no  possibility  of  leaving  some  task  half 
done.  As  has  already  been  emphasized,  it  is  very  desirable 
to  have  sufficient  time  before  the  close  of  the  period  for  a  rapid 
review  and  the  clear  affirmation  in  some  sort  of  summary  of 
the  important  points  of  the  lesson. 

LESSON  VI 

UNIT   OF  INSTRUCTION   II.  —  SEEDS  AND   SEEDLINGS 

LESSON  TYPE.  —  AN  INDUCTIVE  LESSON 

Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 25  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Ask  some  general  review  questions  on  the 
preceding  lesson,  such  as : 

Name  some  dicotyledons,  monocotyledons. 

Where  is  the  food  stored  in  each? 

Of  what  use  is  the  seed  coat? 

Are  all  seeds  alike  ?    Are  all  bean  seeds  alike  ? 

Why  is  it  poor  policy  to  plant  old  seeds  ? 

What  care  should  be  taken  of  seeds  before  they  are  planted? 

A  few  such  questions,  combined  with  a  quick  review  of  the 
parts  of  the  seed  from  the  chart  or  drawing,  will  suffice  to 
cover  the  review. 

The  Assignment.  —  The  new  lesson  will  be  similar  to  the 
preceding  one  in  that  it  will  be  a  continuation  of  the  study  of 
the  textbook,  exemplified  in  every  possible  particular  by  the 
use  of  real  specimens.  There  will  be  also  the  starting  of 
several  experiments  which  from  their  very  nature  must  take 


176    Supervised  Study  in  Mathematics  and  Science 

three  or  four  days  for  their  completion.  These  experiments 
will  be  along  the  line  suggested  by  the  paragraph  on  germina- 
tion in  the  new  lesson.  The  author  states  that  "  when 
supplied  with  moisture,  warmth,  and  oxygen  (air),  it  (the 
embryo)  grows."  1  We  wish  to  verify  these  facts.  We  have 
already  started  an  experiment  to  discover  the  value  of  moisture 
as  a  factor  of  rapid  growth  by  our  planting  of  soaked  and  un- 
soaked  seeds.  We  shall  now  make  another  experiment. 
Tell  the  pupils  that  every  time  we  make  an  experiment,  it 
is  necessary  that  we  have  a  so-called  check  or  control  experi- 
ment, the  one  to  show  that  certain  things  happen  under  certain 
conditions  and  the  other  to  show  that  without  these  conditions 
they  will  not  happen. 

The  pupils  should  be  told  to  open  their  notebooks  and, 
under  separate  heads,  note  what  is  done  to  prove  the  three 
things  stated  in  the  above  quotation. 

Experiment  I:  Do  seeds  need  moisture? 

Procedure:  Let  each  pupil  plant  a  seed,  bean  or  corn,  in  a  plant 
dish.  Designate  two  pupils  to  keep  it  watered.  Tell  all  the  pupils 
to  consider  this  C  in  their  notebooks,  and  so  label  the  dish.  In 
another  dish,  marked  D,  plant  some  of  the  same  seeds,  remarking 
that  this  dish  will  not  be  watered. 

Experiment  II:  Do  seeds  need  warmth? 

Procedure:  Have  the  pupils  plant  seeds  in  two  dishes,  both  to  be 
kept  watered,  one  to  be  set  in  the  sunlight  where  it  is  warm,  the 
other  set  aside  in  some  cool  place.  Label  them  E  and  F  respectively. 

Experiment  III:  Do  seeds  need  oxygen? 

1  Bailey  and  Coleman,  "  First  Course  in  Biology  " ;  The  Macmillan  Company, 
1908. 


Divisions  of  Biology  177 

Procedure:  Make  the  double  planting  of  moistened  seeds  in 
dishes  labeled  G  and  H,  one  to  be  left  uncovered  and  the  other 
securely  corked. 

As  soon  as  the  pupils  have  made  the  requisite  records  in 
their  notebooks,  pass  on  to  the  study  of  the  assignment. 

The  Study  of  the  Assignment.  —  Study  carefully  the 
assigned  pages  in  the  text  which  we  shall  assume  cover  the 
germination  of  seeds.  Although  the  seeds  planted  yesterday 
of  course  have  not  sprouted  yet,  the  teacher  should  have  some 
beans  in  the  various  stages  of  growth  ready  for  study,  having 
planted  them  at  various  intervals  previous  to  this  time.  When 
those  planted  by  the  class  have  begun  to  come  up,  the  class 
should  use  them  for  review  and  for  their  drawings.  They  will 
thus  serve  as  a  verification  of  the  textbook  and  also  as  the 
models  for  the  drawings. 

It  will  be  seen  that  the  teacher  must  be  continually  planning 
for  the  future :  by  having  seeds  at  the  right  stages  of  germina- 
tion; by  having  plants  at  the  right  stages  of  development; 
by  having  plenty  of  illustrative  material  ready  at  the  proper 
time ;  by  keeping  note  of  the  progress  of  various  experiments ; 
and  by  taking  care  that  all  phases  of  the  subject  are  studied 
in  their  right  sequence.  A  great  deal  of  this  may  be  done  by 
the  pupils  themselves,  outside  of  school  hours.  The  teacher  will 
find  that  they  will  be  glad  to  do  this  work.  As  far  as  possible 
pupils  should  be  rotated  in  these  preparations,  so  that  all  may 
feel  a  corresponding  responsibility.  The  biology  laboratory 
should  be  at  all  times  full  of  growing  plants  and  flowers ;  there 
should  be  an  aquarium,  cages  for  living  animals,  herbariums, 
etc.  The  pupils  should  feel  that  their  biology  room  is  es- 
sentially a  place  of  living  things,  and  they  should  be  encouraged 
to  contribute  as  much  material  as  they  can  from  time  to 


178    Supervised  Study  in  Mathematics  and  Science 

time.  The  pupils  should  also  be  encouraged  to  make  similar 
experiments  at  home  and  to  compare  their  results  with  those 
obtained  in  the  laboratory.  Incidentally  it  might  serve  to 
arouse  a  cooperative  interest  in  their  school  work  on  the  part 
of  their  parents.  r 

There  should  also  be  plenty  of  cabinet  room  for  permanent 
specimens.  The  pupils  should  be  encouraged  to  add  to  this 
collection  either  temporarily  or  permanently.  In  this  way  a 
very  adequate  museum  may  be  accumulated  which  may  be 
used  for  demonstration  purposes.  The  teacher  should  em- 
phasize at  all  times,  however,  the  greater  importance  of  living 
things  since  biology  is  essentially  the  study  of  life.  The 
author  has  little  use  for  biological  specimens  purchased  from 
commercial  houses;  some  specimens,  which  are  foreign  to 
the  locality,  must  necessarily  be  purchased,  such  as  star- 
fish, etc.  These,  however,  have  only  a  subordinate  value ; 
the  pupils'  environment  should  furnish  the  mass  of  illustrative 
material.  If  it  is  a  rural  community,  the  specimens  should  be 
those  of  the  farm ;  if  an  urban  community,  those  of  the  locality 
are  to  be  emphasized. 

More  can  be  learned  about  birds  from  some  pupil's  bringing 
his  pet  canary  to  school  or  from  a  trip  to  the  woods  and  study- 
ing the  birds  as  they  are  in  their  natural  habitat,  than  from 
cases  filled  with  mounted  specimens  of  birds,  native  to  foreign 
localities.  At  best,  all  the  latter  can  do  is  to  serve  as  means 
of  identification,  and  it  is  doubtful  if  a  child  will  ever  learn  to 
know  the  birds  except  from  the  study  of  the  live  birds  them- 
selves. Let  us  inculcate  a  love  for  wild  life  in  its  natural 
setting,  even  if  the  specimens  be  limited  in  number ;  at  least, 
let  us  attempt  to  make  biology  a  study  of  what  it  is  generically 
intended  to  be,  that  of  lif  e  and  living  things. 


Divisions  of  Biology  179 

LESSON  VII 

UNIT  OF  INSTRUCTION  H.  —  SEEDS  AND   SEEDLINGS 
LESSON  TYPE.  —  A  SOCIALIZED  LESSON 
Program  or  Time  Schedule 

At  least  60  minutes 

Purpose.  —  The  purpose  of  this  lesson  is  to  study  nature  in 
its  own  setting,  to  learn  some  of  the  lessons  from  nature's  great 
laboratory,  the  out-of-doors.  Field  excursions  are  absolutely 
essential  to  the  proper  evaluation  of  biology,  and  the  more  of 
them  the  better ;  but  let  them  always  have  some  specific 
object  in  view,  some  particular  lesson  to  be  learned.  Inci- 
dentally many  other  lessons  will  be  learned,  but  some  out- 
standing purpose  for  each  trip  should  govern  the  excursion. 

As  soon  as  the  class  assembles,  explain  that  you  are  going  to 
take  a  field  trip  for  the  main  purpose  of  studying  the  ways  in 
which  seeds  are  scattered  or  dispersed.  Tell  the  pupils  to 
have  their  eyes  open  for  the  observation  of  anything  else  which 
pertains  to  botany,  especially  any  illustrations  that  would 
seem  to  verify  the  lessons  already  learned  in  class.  But  the 
main  object  of  the  expedition  is  to  study  seed  dispersal.  All 
the  pupils  should  have  their  notebooks  handy  and  all  examples 
of  this  phase  of  plant  study  should  be  noted  therein. 

A  few  words  on  the  proper  conduct  of  a  field  trip  may  not 
be  out  of  place.  Foremost,  as  in  all  laboratory  work,  and  this 
is  but  a  phase  of  laboratory  work,  the  teacher  should  actually 
take  the  trip  himself  a  day  or  so  previously  so  that  all  the 
illustrative  material  looked  for  may  be  sure  to  be  found  within 
the  time  allotted.  The  teacher  must  be  fairly  well  acquainted 
with  the  surrounding  country  and  should  know  exactly  where 


i8o    Supervised  Study  in  Mathematics  and  Science 

to  go,  how  to  get  there,  about  the  time  it  will  take,  etc.  If  it 
is  impossible  to  complete  the  trip  during  a  regular  period  then 
some  other  hour  should  be  arranged,  but  it  is  better  if  it  can 
be  completed  within  the  hour  allotted. 

The  discipline  of  a  field  trip  will  never  give  one  any  trouble, 
if  the  trip  is  well  planned  and  if  the  class  realize  that  you  hold 
them  as  accountable  for  their  conduct  as  in  the  regular  class- 
room. Of  course  there  will  be  some  freedom  not  allowed 
ordinarily,  but  there  should  be  no  waste  of  time,  no  wander- 
ing from  the  class,  and  no  boisterousness.  The  pupils  are  out 
for  a  specific  purpose,  and  they  should  be  made  to  realize  that 
you  will  tolerate  no  infraction  of  discipline. 

The  excursion  should  take  one  through  the  fields,  the  woods, 
and  the  pastures.  Many  examples  of  seed  dispersal  may  be 
found,  some  that  have  already  been  noted  in  the  text  and  some 
that  perchance  have  been  omitted.  The  pupils  should  ex- 
change experiences  as  they  go  along;  and  if  one  finds  a  good 
example,  the  others  should  have  their  attention  called  to  it. 
Let  it  be  more  or  less  in  the  form  of  a  game  or  contest  to  see 
who  can  find  the  most  examples.  Many  other  illustrations 
of  their  work  in  botany  will  also  be  observed,  such  as  plant 
societies,  the  struggle  for  life,  etc. 

The  teacher  will  need  to  be  alert  every  minute,  and  he  need 
not  feel  abashed  if  the  pupils  ask  many  questions  which  he  can- 
not answer.  It  would  take  a  very  wise  man  indeed  to  know 
all  the  secrets  of  nature.  It  is  a  much  better  sign  to  have 
pupils  ask  questions  which  you  cannot  answer  than  to  have 
them  ask  no  questions  or  show  no  interest  in  the  trip. 

If  seeds  of  plants  whose  names  you  do  not  know  are  being 
dispersed,  make  a  record  in  the  notebook  and  take  back  speci- 
mens to  the  schoolroom  for  later  identification.  Properly 


Divisions  of  Biology  181 

planned,  the  trip  should  be  full  of  valuable  experiences  and 
should  open  up  a  new  vista  to  the  pupils.  As  you  proceed 
in  the  study  of  this  subject  and  later  expeditions  are  under- 
taken, the  pupils  will  find  an  endless  amount  of  corroborative 
material  for  the  work  covered. 

Possibly  the  first  seeds  being  dispersed  may  be  the  dande- 
lions'. Some  late  specimens  of  this  weed  may  be  found  in  the 
school  yard,  yet  how  many  will  ever  have  sensed  before  the 
meaning  of  the  tiny  hairlike  wings  of  this  seed?  Ask  the 
pupils  to  blow  some  to  see  how  far  the  wind  will  carry  them. 
Require  drawings  of  one  of  the  winged  seeds  to  be  made  in  the 
notebooks.  The  next  specimen  may  be  a  milkweed,  and  as 
the  pupils  get  into  the  open  country  they  may  find  the 
snapdragon,  beggar 's-lice,  etc.  A  cow  may  pass  with  her  tail 
tangled  with  burdocks,  a  tumbleweed  may  blow  merrily 
down  the  field,  scattering  its  seeds  right  and  left,  and  so  on 
indefinitely  may  nature  give  an  illustrated  lecture  of  one  of 
her  many  phases. 

LESSON  VIII 

UNIT   OF  INSTRUCTION   IX. —  INSECTS 

LESSON  TYPE.  —  A  How  TO  STUDY  LESSON 

Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 25  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject.    Crustaceans. 
Method.    Ask  the  questions  found  in  the  text,  supplying 
others  of  a  like  nature,  such  as : 


182    Supervised  Study  in  Mathematics  and  Science 

Name  some  fresh-water  crustaceans. 

Name  some  salt-water  crustaceans. 

Name  some  crustaceans  used  as  food. 

What  is  the  general  weapon  of  defense  of  all  crustaceans? 

Then  call  on  someone  to  go  to  the  chart  and  point  out  and 
name  the  various  parts  of  the  crayfish. 

The  Assignment.  —  Tell  the  pupils  that  we  are  now  about 
to  start  the  study  of  another  arthropod,  the  insect.  Tell  them 
we  shall  take  as  our  type  form  the  grasshopper,  since  our 
study  of  it  will  answer  anatomically  for  all  insects.  Ask  some- 
one to  state  the  prominent  characteristics  of  crustaceans. 
Then  have  the  pupils  open  their  books  and  call  on  someone  to 
read  the  characteristics  of  insects.  These  are  found  to  be :  three 
body  divisions  and  six  legs.  They  may  now  be  told  to  close 
their  books  again  while  you  proceed  to  give  them  a  little 
inspirational  preview  on  insects  in  general.  Many  pupils  may 
have  a  predetermined  distaste  or  even  horror  for  bugs  and  this 
will  serve  to  put  them  in  the  proper  attitude  toward  insects 
in  general.  Tell  them  that  all  insects  are  wonderfully  made, 
that  they  have  a  complete,  although  in  many  insects,  a  very 
minute  nervous  system.  This  high-strung  nervous  system  is 
one  of  the  reasons  why  certain  insects,  like  bees,  are  very 
apt  to  resent  and  resent  immediately  any  rapid  movement 
which  may  be  interpreted  by  them  as  indicative  of  harm  to 
them.  Tell  them  a  few  of  the  wonderful  things  about  ants, 
how  they  keep  their  "  cows"  (aphids)  and  milk  them,  how  they 
organize  for  battle,  etc. ;  tell  them  something  about  the  wonder- 
ful life  history  of  the  butterfly ;  tell  them  that  many  scientists 
have  spent  their  lives  studying  the  life  histories  of  various 
insects  and  have  written  some  very  interesting  books  about 
them,  as  did  Romanes,  Lord  Avebury,  Darwin,  Fabre,  and 


Divisions  of  Biology  183 

others.  Tell  them  that  some  insects  are  very  valuable  to 
mankind,  as  the  ladybug,  the  ichneumon  fly,  the  honeybee ; 
and  others  of  course  are  very  harmful,  some  carrying  diseases, 
as  the  mosquito,  some  destroying  clothing,  as  the  clothes  moth, 
and  so  on.  Then  remark  that  in  addition  to  a  specific  study  of 
insects  in  general,  we  shall  spend  some  time  on  the  character- 
istics and  classifications  of  insects,  learn  how  to  mount  them, 
how  to  destroy  the  harmful  ones,  and  how  to  protect  the  useful 
ones.  In  all,  our  study  of  insects  is  likely  to  be  one  of  the 
most  interesting  and  fascinating  phases  of  our  work  in 
zoology. 

Then  ask  the  class  to  name  the  insects  they  know,  at  the 
same  time  writing  their  names  upon  the  board.  This  will 
serve  to  ascertain  just  how  extensively  they  know  insects  and 
will  concentrate  their  interest  at  once  upon  the  subject,  for 
they  will  feel  that  their  present  knowledge  is  of  importance  at 
the  outset. 

The  Study  of  the  Assignment.  —  The  study  of  insects  and 
of  the  grasshopper  in  particular  will  follow  closely  the  sugges- 
tions outlined  in  the  study  of  seeds.  All  pupils  should  supply 
themselves,  if  possible,  with  live  specimens  of  native  grass- 
hoppers. If  it  is  possible  for  the  class  to  go  out  into  the  fields 
and  collect  some  live  specimens,  it  would  be  an  excellent  thing 
to  do,  but  the  season  of  the  year  in  which  this  topic  is  reached 
may  preclude  this  possibility,  in  which  case  the  insects  must  be 
purchased  or  have  been  kept  in  cages  and  raised  for  this  pur- 
pose. With  the  specimen  before  him,  the  pupil  should  ac- 
company the  text  with  the  actual  handling  and  study  of  the 
insect  itself.  Some  textbooks,  as  Bailey  and  Coleman,1  are 
themselves  guides  for  laboratory  work  and,  if  the  teacher  elects, 
1 "  First  Course  in  Biology  " ;  The  Macmillan  Company,  1908. 


184    Supervised  Study  in  Mathematics  and  Science 

may  be  made  the  basis  for  the  notebook  work,  the  pupil  mak- 
ing the  drawings  and  recording  the  facts  learned  from  the 
actual  observation  of  the  grasshopper.  Not  all  the  facts 
observed  should  be  noted  therein,  of  course;  this  might  be 
left  to  the  judgment  of  the  pupil,  or  the  teacher  might  indicate 
on  the  board  certain  facts  which  are  to  be  recorded.  The 
notebook  must  never  be  so  autocratic  as  to  be  petty ;  it 
loses  its  essential  value  unless  it  is  written  up  in  the  expectation 
that  it  shall  serve  as  a  check  on  the  individual  work. 

As  soon  as  the  pupils  have  completed  the  first  two  or  three 
pages  or  that  part  of  the  book  covering  the  external  character- 
istics, the  teacher  should  direct  them  to  close  their  books,  and 
he  should  review  their  work,  asking  questions  corresponding 
to  those  of  the  text.  This  will  help  to  emphasize  the  study, 
will  clarify  any  doubtful  points  or  rectify  any  mistakes  in 
observation,  and  will  further  serve  as  an  opportunity  for  the 
teacher  to  supplement  with  any  additional  data  or  material 
which  he  may  feel  necessary  or  expedient. 

In  the  few  minutes  remaining,  the  teacher  may  unfold  a 
chart  on  the  grasshopper  and  rapidly  review  the  essential 
details  already  studied  regarding  the  external  characteristics 
of  the  insect. 

The  Silent  Study.  —  As  far  as  it  is  possible,  insist  that  the 
pupils  find  out  the  answers  to  the  questions  themselves.  Dis- 
courage their  asking  you  direct  questions.  If  they  make 
incorrect  deductions,  you  can  find  that  out  later,  either  in  the 
quiz  period  or  from  their  notebooks.  The  pupils  should  be 
trained  to  feel  that  they  are  the  ones  to  do  the  work  and  that 
the  teacher's  duty  is  simply  to  direct  that  work,  correct  wrong 
impressions,  and  supplement  their  own  observations.  Facts 
that  the  pupils  ascertain  for  themselves  will  remain  with 


Divisions  of  Biology  185 

them  always,  while  information  you  give  them  will  be  but 
transitory.  The  teacher  should  always  be  ready  to  offer 
suggestions,  however,  to  help  the  pupil  interpret  the  text  if 
he  finds  him  unable  to  do  it  for  himself;  but  the  skillful 
instructor  will  avoid  direct  answers  and  will  strive  to  lead  the 
pupil  to  form  his  own  answer  through  careful  questioning. 
For  instance,  a  pupil  might  call  the  teacher  to  his  side  to  ask 
him  what  is  the  general  shape  of  the  grasshopper.  Instead 
of  saying  outright  that  it  is  cylindrical,  ask  him  what  shape  it 
resembles,  whether  it  is  round  or  square,  circular,  or  rectangu- 
lar, etc.,  whether  it  is  uniform  throughout  or  only  suggests 
some  form  as  a  whole.  If  he  admits  that  it  is  more  or  less 
irregular  but  that  it  is  somewhat  curved,  ask  him  with  what 
sort  of  curved  figure  it  compares  favorably.  In  other 
words,  get  him  to  make  the  final  decision  himself.  If  he 
is  obviously  wrong,  try  by  various  questions  to  get  him  to 
see  that  he  is  wrong  without  directly  telling  him  so.  Re- 
member that  the  pupils  are  learning  judgment  of  fact  as  well 
as  the  facts  themselves.  The  pupil  must  learn  self-reliance, 
a  lesson  much  more  important  than  any  biological  fact  he 
may  acquire.  Indeed  we  may  safely  say  that  all  study 
should  develop  the  powers  of  observation  and  the  training 
of  judgment,  the  ability  to  see  and  to  interpret  correctly 
what  we  see.  It  is  one  great  fault  with  many  teachers  that 
they  often  tell  too  much.  The  teacher  must  know  the  facts, 
not  to  recite  them  but  to  make  sure  that  the  pupils  sense 
them  aright. 

If  the  teacher  finds  that  several  pupils  are  becoming  con- 
fused over  some  question  or  statement,  it  might  be  well  to  take 
the  matter  up  with  the  class  as  a  whole  and  develop  through 
them  the  correct  answer  or  understanding,  just  as  has  been 


1 86    Supervised  Study  in  Mathematics  and  Science 

suggested  before  for  individual  pupils.  However,  the  individ- 
ual method  is  much  to  be  preferred. 

The  Outside  Work.  —  Necessarily  much  of  the  work  in 
biology  must  be  done  during  the  class  period.  The  more  this 
can  be  done,  the  better  will  be  the  results.  Outside  work 
should  consist  largely  of  supplementary  reading,  home  experi- 
ments, field  observations,  special  reports,  tabulations,  col- 
lecting materials,  and  the  like.  Tables  of  comparative  study 
like  that  on  page  85,  Bailey  and  Coleman,  give  excellent  out- 
side work  and  from  their  nature  are  quite  adaptable  for  such 
assignments.  The  biology  library  should  have  a  rich  assort- 
ment of  natural  histories,  bird  books,  flower  guides,  nature 
readers,  etc.,  to  which  the  pupils  may  be  given  assignments 
for  outside  reading  and  reports.  Most  textbooks  furnish 
complete  bibliographies,  and  the  teacher  should  be  reasonably 
acquainted  with  them.  Many  of  the  farm  bulletins  may  be 
used,  such  as  deal  with  the  fly,  the  toad,  raising  bees,  the 
codling  moth,  obnoxious  weeds,  etc. 

An  excellent  plan  is  to  assign  each  day  special  topics  for 
reports  either  written  or  oral,  and  to  make  them  as  varied  as 
possible.  They  should  be  correlated  as  far  as  possible  with 
the  interest  of  the  child,  those  living  on  farms  being  given  some 
topic  relating  more  or  less  to  their  environment.  If  some 
pupil  happens  to  have  chickens,  for  instance,  he  may  be  given 
a  topic :  "  Chicken  Lice  and  How  to  Eradicate  Them  "  ;  or  if 
he  has  a  pet  dog,  give  him  a  pamphlet  on  fleas  and  let  him 
report ;  if  he  is  the  son  of  a  physician,  he  may  be  told  to  look 
up  and  discuss  the  relation  of  the  mosquito  and  malarial  fever, 
and  so  on. 

Encourage  the  pupils  to  bring  to  class  magazine  or  news- 
paper articles  on  some  phase  of  animal  life,  anecdotes  con- 


Divisions  of  Biology  187 

cerning  animal  intelligence,  bird  stories,  etc.  Get  the  pupils 
to  feel  that  their  study  of  biology  is  vital,  practical,  and  full  of 
rich  interest.  They  will  look  through  the  papers  for  inter- 
esting articles,  will  question  other  people  for  experiences,  and 
will  learn  to  observe  and  interpret  many  of  the  interesting 
things  that  are  of  everyday  occurrence. 

LESSON  IX 

UNIT   OF  INSTRUCTION   IX. —INSECTS 

LESSON  TYPE.  —  A  LABORATORY  LESSON 
Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 25  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  The  few  minutes  of  the  review  may  be 
spent  in  the  ordinary  question  method,  quizzing  rapidly 
various  members  of  the  class  on  the  preceding  day's  work. 
The  kind  of  questions  will  depend  on  the  teacher,  but  they 
should  at  least  be  as  thought-producing  as  possible,  although 
the  nature  of  the  subject  necessitates  more  or  less  dogmatic 
answers.  The  best  method  in  asking  fact  questions  is  always 
to  state  the  question  first  and  assign  it  to  some  pupil  after- 
wards. This  intensifies  the  attention  of  all,  as  no  one  knows 
but  he  may  be  called  upon. 

The  Assignment.  —  The  internal  dissection  of  the  grass- 
hopper is  too  complicated  and  minute  to  be  required  of  the 
pupils.  Either  the  teacher  should  make  the  dissection  as  a 
demonstration  or  he  should  use  charts  or  blackboard  drawings 
and  explain  the  general  characteristics  of  the  anatomy  from 


1 88    Supervised  Study  in  Mathematics  and  Science 

them.  We  shall  assume  that  the  charts  will  be  used  and  that 
they  will  resemble  very  closely  those  given  in  the  book. 

Method.  Have  the  pupils  open  their  books  at  the  paragraph 
describing  the  mouth  parts.  Have  someone  who  is  a  good 
reader  stand  and  read  slowly  the  description  of  the  mouth 
parts.  As  he  proceeds,  the  teacher  will  point  them  out  on  the 
chart,  interposing  any  additional  facts  or  explaining  the  text 
in  more  detail  as  he  may  elect.  When  this  has  been  com- 
pleted, ask  some  questions  covering  the  ground  studied. 

Then  pass  on  to  the  next  paragraph  which  may  be  that  on 
respiration,  and  this  may  well  be  covered  in  the  same  way.  In 
this  way  the  new  lesson  on  internal  structure  may  be  gone 
over  entirely,  the  teacher  acting  as  an  interpreter. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assign- 
ment. The  three  or  more  pages  in  the  text  covering  the 
anatomy  of  the  grasshopper. 

II  or  Average  Assignment.  Drawings,  copied,  of  course,  of 
the  mouth  parts,  nervous  system,  and  digestive  system  (in  note- 
book). 

///  or  Maximum  Assignment.  Study  of  two  or  three  micro- 
scopic slides,  such  as  show  the  facets  of  the  eye,  the  veining 
of  the  wings,  and  an  abdominal  segment  containing  a  spiracle. 

The  Silent  Study.  —  With  the  assignment  explained  as  sug- 
gested above,  the  pupils  may  not  require  much  help  during  the 
study  period.  There  may  be  some,  however,  who  even  now  have 
not  thoroughly  grasped  all  that  the  author  has  to  say,  and  the 
teacher  will  be  ready  to  lend  any  additional  aid  if  necessary. 
Particular  attention  should  be  given  to  the  pupils  who  are  not 
doing  satisfactory  work  and  an  effort  made  during  this  period 
to  aid  them.  This  may  take  the  method  of  sitting  down  be- 
side such  a  pupil  and  giving  him  personal  supervision  in  his 


Divisions  of  Biology  189 

study.  Let  him  read  a  few  sentences  and  then  ask  him  ques- 
tions on  it,  thus  making  sure  that  he  is  getting  a  thorough 
understanding  of  the  meaning.  Too  much  emphasis  cannot 
be  placed  on  the  importance  of  the  teacher's  selecting  each  day 
someone  who  is  falling  behind  in  his  work  and  through  a  study 
of  his  method  of  study  trying  to  put  him  on  his  feet.  If  the 
teacher  will  do  this  at  every  opportunity,  he  will  be  well  repaid. 
It  is  trite  to  say  that  the  teacher  should  never  feel  satisfied 
with  his  work  until  he  has  exhausted  every  possible  avenue  of 
assistance  to  enable  a  pupil  to  master  his  work.  Indeed  it 
seems  to  the  writer  that  more  real  good  can  be  done  in  these 
individual  "  first  aid "  sections  than  in  the  regular  class  reci- 
tation periods.  It  is  our  business  as  teachers  to  pay  partic- 
ular attention  to  the  extremes  of  our  classes,  those  doing 
minimum  and  those  doing  maximum  work. 

LESSON  X 

UNIT   OF  INSTRUCTION   IX.  —  INSECTS 

LESSON  TYPE.  —  A  CORRELATION  AND  RESEARCH  LESSON 
Program  or  Time  Schedule 

The  Review 20  minutes 

The  Assignment 15  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  The  object  of  this  review  is  to  emphasize 
graphically  the  entire  order  of  insects  as  to  (a)  general  charac- 
teristics common  to  all  insects,  (&)  outstanding  characteristics* 
of  each  order,  i.e.  wings,  (c)  classification  according  to  mouth 
parts,  i.e.  biting  or  sucking  insects,  and  (<T)  economic  impor- 
tance, i.e.  harmful  or  beneficial. 


190     Supervised  Study  in  Mathematics  and  Science 

Method.  Write  in  yellow  crayon,  at  the  top  of  each  of  four 
panels  of  the  blackboard,  one  of  the  four  suggested  topics. 
When  the  class  assembles,  the  teacher  should  develop  through 
questioning  the  data  which  he  will  write  under  each  caption. 
When  completed,  the  four  panels  should  be  a  correlated  re- 
view of  the  essential  characteristics  of  insects  as  a  class,  and 
it  will  serve  forcibly  to  impress  upon  the  pupils  the  integrity 
of  the  work  on  the  study  of  insects. 

Under  the  first  caption,  ask  someone  to  name  the  character- 
istics which  all  insects  have  in  common.  If  the  pupil  should 
mention  some  point  which  is  not  common  to  all,  instead  of 
ruling  it  out  point-blank,  mention  some  insect  which  does  not 
possess  it  and  thus  direct  the  pupil  how  to  draw  upon  his  own 
knowledge  before  he  answers.  If  the  pupil  should  state  cor- 
rectly the  two  characteristics  desired,  i.e.  three  pairs  of  legs 
and  three  body  divisions,  mention  some  others  and  ask  whether 
they  should  be  allowed.  Exhaust,  in  other  words,  all  possible 
items  that  might  be  included,  making  clear  that  these 
are  the  only  ones  that  may  be  considered  in  this  general 
summary. 

Under  the  next  caption,  write  the  suffix  -ptera  ten  times, 
calling  on  various  pupils  to  supply  the  prefix  and  to  state  the 
meaning,  as  diptera,  two  wings ,  coleoptera,  sheath  wings ;  etc. 
Also  call  for  an  example  under  each  order. 

Under  the  third  caption,  divide  the  board  into  two  sections, 
one  with  the  heading  Biting  Insects  and  the  other  Sticking 
Insects.  Call  on  someone  to  name  all  those  given  under  the 
preceding  classification,  which  may  be  placed  in  the  first  group 
and  those  which  may  fall  into  the  second  group ;  or  possibly  a 
better  way  would  be  to  state  each  one  separately  and  call  on 
someone  to  tell  in  which  group  it  should  be  placed.  If  a  pupil 


Divisions  of  Biology  191 

makes  a  wrong  classification,  instead  of  simply  declaring  it 
incorrect,  direct  him  to  judge  for  himself  whether  he  is  right 
or  wrong.  A  few  questions  concerning  the  characteristics 
illustrative  of  this  order,  which  have  been  noted  on  the  second 
board,  will  help  him  to  do  this. 

On  the  fourth  board,  ask  first  for  insects  which  are  bene- 
ficial, and  insist  in  each  case  that  some  simple  explanation  be 
given  to  characterize  its  value,  as  Bees:  make  honey;  Ich- 
neumon fly:  destroys  harmful  larva  of  deep-boring  insects  through 
deposition  of  its  eggs  in  their  tunnels.  Then  under  harmful 
insects,  do  the  same.  Make  the  questions  terse,  scatter  them 
among  the  pupils,  keep  all  on  the  alert  for  illustrations  and 
for  proper  classifications. 

The  Assignment.  —  As  soon  as  the  four  panels  have  been 
completed,  turn  to  the  new  work,  which  is  a  research  lesson. 
Explain  that  you  are  going  to  give  each  pupil  a  card  which  will 
have  written  upon  it  a  problem  concerning  some  insect,  and 
that  the  book  or  bulletin  containing  the  information  will  be 
found  to  be  indicated  upon  the  same  card.  Direct  them  to 
study  the  problem,  and  then  search  for  its  answer  in  the  ref- 
erences given.  Tell  them,  when  they  are  satisfied  that  they 
have  found  the  correct  answer,  to  write  upon  a  sheet  of  paper 
the  statement  of  the  problem,  its  answer  as  they  have  decided 
it  should  be  given,  and  then  the  reference  by  title  and  page. 
These  papers  may  be  collected  at  the  close  of  the  period  or 
handed  in  the  next  day  if  the  period  does  not  suffice  for  its 
completion. 

The  Study  of  the  Assignment.  —  In  this  case  there  will  be 
only  one  general  assignment,  but  the  questions  may  be  so 
graded  that  the  harder  problems  may  be  given  to  those  show- 
ing more  marked  ability,  and  so  on  down  to  the  easier  ones 


19  2    Supervised  Study  in  Mathematics  and  Science 

which  are  assigned  to  those  who  ordinarily  are  able  to  do  but 
the  minimum  amount. 

Below  is  a  list  of  suggested  topics  and  references : 

1.  When  should  apple  trees  be  sprayed  to  kill  the  codling  moth, 

and  what  is  used  ?    See  "  The  Control  of  the  Codling  Moth  " ; 
Farmers'  Bulletin  No.  171. 

2.  How  did  the  cotton-boll  weevil  get  into  the  United  States? 

See  Weed's  "  Farm  Friends  and  Farm  Foes" ;  D.  C.  Heath 
and  Co. 

3.  What  is  the  estimated  value  of  a  toad  to  the  farmer?    See 

"  The  Usefulness  of  the  Toad" ;  Farmers'  Bulletin  No.  196. 

4.  What  economic  value  has  the  cochineal  bug  ?    See  any  encyclo- 

pedia. 

5.  What  is  parthenogenesis?     See  Bigelow's  "  Applied  Biology"; 

The  Macmillan  Company. 

6.  Why  is  the  tachina  fly  beneficial?     See  Smallwood-Reverly- 

Bailey's  "  Biology  for  High  Schools"  ;  Allyn  and  Bacon. 

7.  What  is  Bordeaux  mixture  and  for  what  is  it  used  ?    See  p.  263, 

Warren's    "Elements   of    Agriculture";     The    Macmillan 
Company. 

8.  How  is  the  Rocky  Mountain  locust  destroyed?    See   p.  15, 

Linville  and  Kelly's  "  Textbook  in  Zoology"  ;  Ginn  and  Co. 

9.  Where  do  flies  spend  the  winter?    See  p.  81,  Hegner's  "  Prac- 

tical Zoology"  ;  The  Macmillan  Company. 

10.   What  is  the  estimated  yearly  economic  loss  from  insects  ?    See 
Hunter's  "Essentials  of  Biology";    American  Book^Co. 

LESSON  XI 
UNIT  OF  INSTRUCTION  IX.  —  INSECTS 

LESSON  TYPE.  —  A  SOCIALIZED  LESSON 
Program  or  Time  Schedule 

The  Review 25  minutes 

The  Assignment 10  minutes 

The  Study  of  the  Assignment 25  minutes 


Divisions  of  Biology  193 

The  Review.  —  Subject  Matter.  Harmful  insects  and  how 
to  get  rid  of  them. 

Method.  Previous  to  the  assembling  of  the  class,  write  upon 
the  board  the  names  of  as  many  harmful  insects  as  there  are 
members  in  the  class.  Such  a  list  would  probably  include  the 
mosquito,  house  fly,  codling  moth,  carpet  moth,  bedbug, 
cotton-boll  weevil,  potato  bug,  squash  bug,  tent  caterpillar, 
etc.  Also  write  upon  separate  slips  of  paper  the  names  of  the 
various  members  of  the  class,  and  have  these  in  a  box. 

Explain  to  the  class  that  they  are  going  to  take  part  in  a  sort 
of  game  which  might  be  called,  "  Ridding  the  Community  of 
Obnoxious  Insects."  Each  pupil  will  be  called  upon  to  tell 
why  the  insect  assigned  to  him  is  harmful  and  how  to  get  rid 
of  it.  Each  time  the  pupil  answers  correctly  concerning  the 
insect  assigned  to  him,  the  name  of  that  insect  will  be  erased 
from  the  board.  The  object  will  be  to  try  to  erase  from  the 
board  or  the  community  all  of  the  names,  i.e.  the  pests. 

Allow  a  few  minutes  for  the  pupils  to  look  up  in  their  text- 
book or  any  other  books  available  any  of  the  insects  mentioned 
on  the  board  on  which  they  wish  to  be  better  posted.  Then 
at  the  end  of  ten  minutes,  call  someone  to  the  front  of  the 
room,  who  might  be  designated  class  entomologist  for  the  nonce, 
and  direct  him  to  draw  some  slip  from  the  box  and  read  the 
name  upon  it.  The  pupil  whose  name  is  read  must  rise  and 
without  help  tell  why  this  particular  insect,  taking  it  in  the 
order  in  which  it  has  been  written  upon  the  board,  is 
harmful  and  how  it  may  be  destroyed.  If  he  can  recite  suc- 
cessfully, the  name  of  that  insect  is  erased ;  otherwise,  it  is  left 
on  the  board,  and  as  a  penalty  for  failing,  he  may  be  required 
to  look  it  up  and  hand  in  a  written  answer  before  he  leaves. 
The  process  is  repeated  until  all  the  pupils  have  been  called 


1 94    Supervised  Study  in  Mathematics  and  Science 

upon,  and  the  board  has  been  cleaned  more  or  less  completely, 
according  to  the  successful  answers. 

A  little  more  spirit  may  be  injected  into  this  game  if  sides 
are  chosen  and  a  contest  developed  to  see  which  side  can 
eradicate  the  larger  number  of  the  harmful  insects. 

The  Assignment.  —  Tell  the  class  that  the  assignment  for 
the  next  day  will  be  in  the  nature  of  a  series  of  short  illustrated 
lectures  by  the  various  members  of  the  class.  Tell  them  they 
are  to  bring  to  class  some  insect  which  they  have  found  on 
some  of  their  field  trips,  or  in  lieu  of  that,  the  picture  of  some 
insect,  and  that  you  will  ask  them,  when  called  upon,  to  come 
forward  and,  holding  the  insect  or  picture  in  their  hands,  de- 
scribe to  the  class  its  characteristics ,  habits,  order,  etc.  Also, 
that  other  members  of  the  class  may  question  them  further 
concerning  their  specimens  and  that  you  will  expect  them  to 
be  posted  fully  on  each  particular  insect.  Suggest  that  they 
follow  the  outline  below  in  describing  their  specimen : 

Name: 
Order : 
Characteristics  of  form : 

"  life  history: 

"  habitat: 
Food: 
Economic  importance : 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 
Each  pupil  to  be  prepared  to  describe  according  to  the  outline 
given  above,  some  insect  to  be  selected  by  himself. 

II  or  Average  Assignment.  Each  pupil  to  bring  to  class  a 
specimen  of  the  insect  he  is  to  describe,  or  at  least  a  picture  of  it. 

777  or  Maximum  Assignment.  Be  able  to  quote  some 
source  of  information  other  than  the  textbook  in  use. 


Divisions  of  Biology  195 

LESSON  XII 
A  RED  LETTER  LESSON  IN  ZOOLOGY 

Time  Schedule 
Program 60  minutes 

Purpose.  —  The  object  of  a  red  letter  lesson  is  to  review 
vividly  all  the  important  phases  of  the  work  covered  in  zoology 
and  also  to  introduce  an  incentive  to  superior  work  during  the 
weeks  devoted  to  the  study  of  some  particular  section  of  the 
work  in  biology.  It  is  recommended  that  at  least  one  such 
lesson  follow  the  completion  of  the  work  in  botany,  zoology, 
and  physiology. 

Method.  The  program  should  be  outlined  some  time  in 
advance,  so  that  all  may  be  making  their  plans  for  this  Grand 
Review. 

The  program  may  be  divided  into  three  sections,  the  first  or 
major  part  of  the  program  being  devoted  to  a  review  of  the 
various  orders  of  zoology,  the  other  two  sections  of  the  program 
being  a  display  of  notebooks,  drawings,  charts,  etc.  Printed 
or  mimeographed  programs  will  add  to  the  enjoyment  of  the 
occasion  and  will  serve  as  a  souvenir,  something  which  young 
people  always  treasure. 

PROGRAM 

SECTION  ONE:    GRAND  REVIEW 

A  five  minute  talk  on  each  of  the  type  forms  studied  during  the 
work  in  zoology : 

The  Paramecium  The  Frog 

The  Crayfish  The  English  Sparrow 

The  Grasshopper  The  Rabbit 

The  Perch 


196    Supervised  Study  in  Mathematics  and  Science 

SECTION  Two:  DISPLAY 

1.  Two  or  three  of  the  best  notebooks. 

2.  Two  or  three  of  the  best  written  descriptions  of  some  bird, 
in  prepared  booklets. 

3.  A  display  of  all  the  specimens  which  have  been  collected 
during  the  year. 

4.  Charts,  done  in  India  ink,  showing  the  fly  nuisance. 

5.  Bird  houses,  flytraps,  animal  snapshots  taken  by  the  pupils, 
etc. 

SECTION  THREE:  ILLUSTRATED  LECTURE 

A  stereopticon  lecture,  showing  slides  of  birds  common  to  the 
neighborhood,  by  the  instructor. 

Procedure.  If  there  are  pupils  who  are  especially  gifted  in 
the  ability  to  draw,  let  them  the  day  prior  to  the  rendition  of 
the  program,  make  drawings  of  the  animals  suggested  in 
section  one,  on  the  blackboard,  using  colored  crayon  if  they 
choose.  These  of  course  may  be  prepared  by  the  instructor 
if  preferred,  or  charts  may  be  utilized  for  this  purpose.  Every 
well-equipped  biology  room  will  necessarily  be  supplied  with 
these  charts,  which  should  be  on  display  at  all  times  on  the 
front  wall  of  the  room.  The  pupils  designated  to  give  these 
talks  should  be  selected  beforehand  and  they  should  have 
their  reports  well  planned. 

The  objects  for  display  in  section  two  should  be  selected  in 
advance,  and  some  pains  taken  to  exhibit  them  in  an  attractive 
way.  The  attention  of  the  class  will  have  been  drawn  to  this 
red  letter  lesson  some  time  before  the  actual  date,  and  the 
pupils  encouraged  to  prepare  something  for  it.  Some  of  the 
charts  on  the  fly  nuisance,  for  instance,  may  easily  be  arranged 
with  the  teacher  of  drawing  and  may  take  the  forms  sug- 
gested in  a  little  pamphlet  issued  by  the  International 


Divisions  of  Biology  197 

Harvester  Company  of  New  Jersey,  entitled,  "  Trap  the 
Fly." 

Pupils  in  the  manual  training  department  may  also  be  en- 
couraged to  make  bird  houses,  flytraps,  etc.,  or  these  may  be 
borrowed  from  some  of  the  pupils  in  the  grades  where  this 
work  is  done  in  connection  with  nature  study.  Some  very 
interesting  kodak  pictures  of  birds,  mammals,  and  other 
animals  will  be  forthcoming  from  the  announcement  of  this 
feature.  It  may  also  serve  the  double  purpose  of  interesting 
the  pupils  in  that  sport  of  hunting  with  the  camera  which  is 
so  much  more  to  be  commended  than  hunting  with  a  gun. 

A  short  illustrated  lecture  on  birds  will  also  be  easily  ar- 
ranged if  the  school  is  supplied  with  a  lantern.  In  New  York 
State  slides  are  furnished  without  charge  by  the  Division  of 
Visual  Instruction,  and  in  case  they  are  not  accessible  from 
some  free  source,  there  are  a  number  of  firms  which  will  rent 
or  sell  suitable  slides  very  reasonably. 

LESSON  XIII 

UNIT  OF  INSTRUCTION   XVI.  —  BONES  AND   MUSCLES 
LESSON  TYPE.  —  A  How  TO  STUDY  LESSON 
Program  or  Time  Schedule 

The  Review 15  minutes 

The  Assignment 20  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Subject  Matter.    Bones. 

Method.  Let  one  of  the  pupils  go  to  the  skeleton  or  manikin 
and  point  to  the  various  bones  and  call  on  individual  members 
of  the  class  to  describe  them  as  to  name,  kind,  and  use,.  He 


198    Supervised  Study  in  Mathematics  and  Science 

is  to  be  the  judge  of  the  correctness  of  the  recitation  given ;  and 
if  he  accepts  an  erroneous  answer,  let  him  take  his  seat  and 
assign  another  to  act  as  temporary  teacher.  Pupils  like 
occasionally  to  have  the  responsibility  of  playing  teacher  and 
it  is  excellent  practice,  calling  as  it  does  for  accurate  knowl- 
edge and  for  display  of  judgment. 

The  Assignment.  —  Have  some  pupil  step  to  the  front  of 
the  room,  open  his  book  at  the  chapter  on  muscles,  and  read 
the  first  paragraph,  the  rest  of  the  class  meantime  reading  with 
him  silently.  Ask  the  one  who  read  the  paragraph  if  there  are 
any  statements  of  the  author  he  does  not  understand.  If 
there  are,  either  bring  out  the  meaning  through  careful  ques- 
tioning or  explain  it  in  simple  language.  If  there  are  any 
words  which  are  new  and  are  likely  to  give  trouble,  have  some- 
one look  them  up  in  the  dictionary  and  report.  Now  have 
all  the  pupils  close  their  books,  and  ask  the  one  who  has  been 
reading  to  state  the  substance  of  the  paragraph  in  his  own 
words.  Call  on  one  or  two  others  to  do  likewise.  Then  pro- 
ceed in  a  similar  manner  with  the  next  paragraph.  In  case 
the  paragraph  has  necessitated  quite  a  bit  of  explaining,  it 
might  be  well  to  have  the  pupils  reread  it  very  carefully 
before  reproducing  it.  In  this  way  the  pupils  will  be  taught 
how  they  should  study  their  new  assignment. 

The  principal  object  of  this  intensive  study  of  the  assignment 
will  be  to  inculcate  a  method  of  study  which  shall  be  at  once 
exhaustive,  intelligent,  and  comprehensive.  Too  much  so- 
called  study  is  simply  mechanical  reading  of  the  printed  page 
without  assimilating  what  the  author  has  to  say.  The  habit 
of  retrospection  of  each  paragraph  before  proceeding  to  the 
next  is  an  invaluable  acquisition  to  the  pupil  who  desires  to 
master  the  subject  matter. 


Divisions  of  Biology  199 

The  Study  of  the  Assignment.  —  /  or  Minimum  Assign- 
ment. 

Four  or  five  pages  of  the  text  in  the  new  chapter  which  deals 
with  muscles. 

II  or  Average  Assignment. 

Draw  diagrammatically  a  group  of  involuntary  muscle 
cells. 

III  or  Maximum  Assignment. 

The  following  or  similar  thought  questions : 

a.  Why  is  it  important  that  some  muscles  are  voluntary? 
Name  two  or  three. 

6.  Mention  some  involuntary  muscle  which  may  be  made 
voluntary  at  the  desire  of  the  owner. 

c.   What  makes  a  muscle  red ?   tough?  elastic? 

The  Maximum  Assignment.  —  When  the  maximum  assign- 
ment consists  of  auxiliary  thought  questions,  they  should  be 
questions  that  are  not  treated  in  the  textbook  in  use,  but 
should  either  be  of  such  a  nature  as  to  be  inferred  with  a  little 
thought  or  to  necessitate  the  use  of  other  books.  Usually  the 
answers  should  be  carefully  written  out  and  handed  in  the 
next  day.  If  other  books  are  used  for  authority,  their  titles  and 
the  names  of  the  authors  should  be  stated  as  references.  It 
should  be  made  emphatic  that  answers  should  be  authorita- 
tive and  should  be  substantiated  by  concrete  references  to  the 
authorities.  If  this  method  becomes  habituated,  many  loose 
statements  common  to-day  will  in  time  be  done  away  with,  for 
as  the  child  learns  to  act  and  think  and  make  statements 
during  his  school  days  when  his  mind  is  being  trained  to  react 
along  definite  lines,  these  habits  are  likely  to  become  the  out- 
standing characteristic  of  his  more  mature  attitude  as  a 
student. 


2oo    Supervised  Study  in  Mathematics  and  Science 
LESSON  XIV 

UNIT   OF  INSTRUCTION   XVI.  —  MUSCLES 

LESSON  TYPE.  —  A  LABORATORY  LESSON 
Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 10  minutes 

The  Study  of  the  Assignment 40  minutes 

The  Review.  —  Picking  out  some  of  the  important  things  to 
be  reviewed,  quiz  the  class  rapidly  and  intensively  on  the 
things  deemed  essential.  Many  of  the  questions  ordinarily 
asked  during  the  recitation  period  may  well  be  dispensed  with 
since  they  are  of  minor  importance  and  have  presumably  been 
drilled  on  sufficiently  during  the  regular  work  on  this  chapter. 
This  short  review  may  be  more  in  the  light  of  summarizing 
some  of  the  more  essential  facts  about  muscles.  Ten  minutes, 
used  intensively,  and  thoroughly  planned  as  to  the  objects 
desired,  may  well  suffice  for  to-day  as  the  laboratory  work 
will  take  the  greater  part  of  the  period. 

The  Assignment.  —  Explain  that  the  work  to-day  will  be 
the  study  of  a  few  microscopic  slides.  Have  the  slides  that 
you  wish  examined  carefully  selected  and  arranged  before- 
hand so  that  there  may  be  no  loss  of  time.  The  compound 
microscope  should  be  adjusted  and  the  first  slide  to  be  ex- 
amined in  place.  A  large  diagrammatic  drawing  of  what  will 
be  seen  under  the  microscope  should  also  have  been  drawn 
upon  the  board  and  the  attention  of  the  class  called  to  the 
essential  things  that  the  pupils  are  to  try  to  see  in  the  specimen. 
While  the  class  is  examining  the  drawing  on  the  board,  call 
someone  to  the  microscope  and  have  him  examine  the  slide, 


Divisions  of  Biology  201 

referring  occasionally  to  the  blackboard  drawing  for  com- 
parison and  verification.  Then  let  him  return  to  his  seat, 
copy  the  drawing  on  the  board  and  note  under  it  any  remarks 
that  seem  to  be  necessitated  by  his  verified  examination  of  the 
slide.  As  soon  as  the  first  pupil  has  used  the  microscope, 
have  another  pupil  ready  to  take  his  turn.  In  this  way,  with 
a  little  careful  executive  ability,  the  entire  class  will  be  enabled 
to  examine  a  half  dozen  or  more  slides. 

It  is  here  assumed  that  the  pupil  has  already  had  experience 
with  the  compound  microscope,  but  if  not,  then  a  few  minutes 
should  be  taken  at  the  outset  to  explain  the  workings  of  the 
instrument  and  to  give  directions  as  to  how  one  should  look 
through  it.  Do  not  allow  a  pupil  to  make  any  adjustments 
of  the  instrument  as  the  inexperienced  are  not  to  be  trusted  to 
turn  the  adjustment  screws.  The  slides  may  easily  be  broken, 
or  the  objective,  and  adjustment  is  too  complicated  a  pro- 
ceeding for  the  ordinary  pupil  to  attempt.  If  he  finds  that 
his  eyes  are  such  as  to  require  a  readjustment,  the  teacher 
should  always  give  the  needed  assistance.  At  some  other 
time,  if  the  instructor  sees  fit,  individual  instruction  in  adjust- 
ment may  be  given  to  the  pupils  but  always  with  slides  made 
for  the  occasion  and  not  with  those  forming  the  equipment  of 
the  biological  laboratory. 

On  Previous  Arrangement  of  Blackboard  Material.  It  will 
possibly  have  been  noted  in  these  illustrative  lessons  that 
much  emphasis  has  been  placed  on  the  diagrams  and  data  to 
be  placed  on  the  blackboard  by  the  instructor  previous  to  the 
assembling  of  the  class.  The  science  teacher  above  all  else 
should  be  able  to  make  good  diagrammatic  drawings.  The 
use  of  the  board  should  be  almost  wholly  employed  by  the 
teacher.  Colored  crayons  do  much  to  emphasize  the  essential 


2O2    Supervised  Study  in  Mathematics  and  Science 

features  of  the  drawings,  which  should  be  carefully  executed 
and  labeled;  they  should  also  be  fairly  large.  The  good 
teacher  is  a  good  executive  and  one  of  the  first  rules  of  exec- 
utive ability  is  careful  planning,  with  strict  attention  to  all 
details  and  possible  complications. 

The  Study  of  the  Assignment.  —  Give  as  a  general  assign- 
ment for  all  a  study  of  the  drawings  the  pupils  have  made  and 
the  verification  of  their  authenticity  through  references  to 
other  books  which  give  illustrations  of  similar  drawings.  Have 
them  look  carefully  to  see  whether  their  drawings  compare 
favorably  with  these  book  drawings  and  if  not,  to  find  out  why. 

Some  of  the  slides  recommended  for  this  lesson  are : 

a.  Involuntary  muscle  cells  or  fibers. 

b.  Voluntary  muscle  cells  or  fibers. 

c.  Heart  muscle  cells. 

d.  Motor  nerve  fibers  ending  among  fibrils  of  voluntary  muscle. 

e.  Capillaries  among  fibers  of  voluntary  muscle. 

LESSON  XV 

UNIT  OF  INSTRUCTION   XVI.  —  MUSCLES 
LESSON  TYPE.  —  A  DEDUCTIVE  LESSON 

Program  or  Time  Schedule 

The  Review 20  minutes 

The  Assignment 15  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Assign  to  as  many  pupils  as  you  had  differ- 
ent slides  under  examination  the  preceding  day  the  task  of 
going  to  the  board  and  making  from  memory  a  rough  sketch 
of  the  slide  assigned.  While  they  are  at  the  board,  quiz  the 
class  on  the  essential  features  which  they  found  in  their  mi- 


Divisions  of  Biology  203 

croscopic  study  of  these  slides.  When  the  drawings  have  been 
completed,  ask  the  class  for  criticisms.  If  any  are  made  which 
are  well  justified,  have  the  pupil  making  the  proper  criticism 
go  to  the  board  and  make  the  alterations. 

The  Assignment.  —  Tell  the  class  that  the  work  to-day  will 
be  in  the  nature  of  the  problem :  How  does  muscular  activity 
aid  the  health  of  the  individual?  The  class  will  agree  that 
muscular  activity  does  improve  one's  health,  but  the  question 
is — just  how?  All  textbooks  will  have  something  on  this 
topic,  but  none  will  deal  with  it  exhaustively  and  the 
teacher  will  be  able  to  make  definite  references  to  various 
authorities  for  more  advanced  research.  Encourage  the  pupils 
to  add  to  the  references  noted  by  discovering  others. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 

a.  Cite  cases  where  lack  of  muscular  activity  has  resulted  in 
poor  health. 

Reference:  p.  48,  Bailey  and  Coleman,  "  First  Course 
in  Biology."1 

b.  Cite  cases  where  regular  muscular  activity  has  resulted  in 
improved  health. 

Reference:  Chapters  IV  and  V,  O'Shea  and  Kellogg, 
"  Making  the  Most  of  Life."  1 

c.  Mention  some  muscular  activities  which  might  be  good 
for  improving  the  health. 

Reference:      Chapter     VH,     O'Shea     and     Kellogg, 
"  Health  Habits."  1 
II  or  Average  Assignment, 
a.  What  becomes  of  muscles  which  are  not  exercised? 

Reference :  Chapter  II,  Jewett,  "  The  Body  and  Its 
Defenses."  2 

1  The  Macmillan  Company.  *  Ginn  and  Co. 


2O4    Supervised  Study  in  Mathematics  and  Science 

b.  What  happens  to  muscles  when  they  are  exercised? 
Reference:   Chapter  III,  Jewett,  "The  Body  and  Its 

Defenses." 1 

c.  Explain  the  biological  result  of  exercise  on  muscles. 
Reference :  Lagrange,  "  Physiology  of  Bodily  Exercise."  z 

III  or  Maximum  Assignment. 

a.  Discuss  the  following  exercises  as  to  their  specific  value : 
walking,  swimming,  chopping  wood,  playing  tennis,  setting- 
up  exercises,  use  of  gymnastic  apparatus. 

Reference :  Chapter  XXII,  Hutchinson,  "  Handbook  of 
Health."3 

b.  List  some  cautions  to  keep  in  mind  while  exercising. 
Reference :  Chapter  XIX,  Eddy,  "  Textbook  in  General 

Physiology  and  Anatomy." 4 

c.  Discuss :  A  healthy  mind  needs  a  heal  thy  body. 
Reference:  Lee,  "Play  in  Education"5;   Groos,  "The 

Play  of  Man."  2 

LESSON  XVI 

UNIT  OF  INSTRUCTION  XVI.  —  MUSCLES 

LESSON  TYPE.  —  A  LESSON  IN  CORRELATION 
Program  or  Time  Schedule 

The  Review 20  minutes 

Program 40  minutes 

The  Review.  —  Call  on  different  pupils  to  stand  and  tell  all 
they  can  to  substantiate  the  truth  of  the  proposition  that 
muscular  activity  tends  to  good  health.  Let  each  recitation 

1  Ginn  and  Co.        2  D.  Appleton  and  Co.        3  Houghton  Mifflin  Co. 
4  American  Book  Co.        B  The  Macmillan  Company. 


Divisions  of  Biology  205 

be  as  complete  as  possible,  being  in  the  nature  of  an  exposition 
of  this  particular  problem.  At  the  conclusion  of  each  recita- 
tion, make  any  suggestions  that  will  tend  to  direct  the  next 
pupil  to  make  his  contribution  logical  and  clear.  Make  the 
review  not  only  scientific  in  nature  but  an  exposition  in  good 
English,  thus  directly  correlating  the  work  with  the  oral 
English.  The  teacher  should  never  accept  as  final  any  ex- 
tended recitation  which  does  not  follow  the  lines  of  correct 
English,  logical  sequence  of  thought  and  well-rounded  sen- 
tences. Habituate  the  pupils  in  the  use  of  short  sentences, 
pure  English,  and  definite  statements.  Explain  that  the  mere 
reciting  of  scientific  data  without  regard  to  its  recital  in  the 
best  structural  form  causes  the  loss  of  much  of  its  inherent 
value.  It  is  a  pet  theory  of  the  author  that  aside  from  a 
short  course  in  technical  English,  the  place  to  teach  rhetoric 
and  English  composition  efficiently  is  in  the  various  subjects 
of  the  school  program.  The  teacher  should  feel  that  primarily 
he  is  to  develop  the  power  of  the  pupil  to  express  himself  and 
to  sense  that  science,  history,  etc.  are  important  largely  for 
the  fact  that  they  supply  the  pupil  with  the  material  for  his 
conversation  and  composition.  A  man  may  be  very  proficient 
in  his  knowledge  of  bird  life,  but  if  he  is  unable  to  express  him- 
self in  a  clear,  logical,  and  pleasing  manner,  his  knowledge  is 
apt  to  be  of  but  little  use  to  himself  or  to  others.  Teachers 
are  too  often  prone  to  be  satisfied  with  the  mere  acquisition  of 
facts  and  to  pass  over  as  of  little  or  slight  importance  the 
exposition  of  those  facts. 

The  Assignment.  —  The  Object  of  a  Lesson  in  Correlation. 
All  school  work  should  have  more  or  less  interrelationship. 
The  correlation  of  English  and  the  work  in  biology  has  just 
been  noted.  The  work  of  muscles  and  their  development 


206    Supervised  Study  in  Mathematics  and  Science 

through  athletic  and  other  activities  have  been  for  the  past  few 
lessons  the  object  of  investigation.  Systematic  physical  train- 
ing has  been  mentioned  as  one  of  the  best  means  of  developing 
the  muscles  and  thus  improving  the  health  of  the  individual. 
Explain  that  you  have  asked  the  physical  instructor  to  give 
the  class  a  series  of  exercises,  with  proper  explanations,  which 
will  tend  to  develop  the  various  muscles  of  the  body.  Ask 
them  to  take  notes  on  the  various  exercises  and  to  be  able  to 
demonstrate  each  to-morrow,  with  a  few  words  regarding  their 
special  functions. 

The  physical  instructor  may  now  be  introduced,  and  he  will 
proceed  to  demonstrate  various  setting-up  exercises  which 
have  as  their  essential  object  the  strengthening  of  various  sets 
of  muscles.  If  the  plan  of  the  demonstration  has  been  talked 
over  with  the  physical  instructor  prior  to  the  class  period,  a 
very  helpful  and  interesting  lesson  will  result.  The  physical 
instructor  on  his  side  will  welcome  this  opportunity  to  demon- 
strate the  scientific  basis  for  his  work,  and  the  pupils  will  come 
to  have  a  new  realization  of  the  specific  value  of  the  setting-up 
exercises  which  they  have  been  doing  from  day  to  day,  and 
physical  training  will  have  a  new  meaning  for  them.  The 
value  of  the  demonstration  will  depend,  of  course,  upon  the 
exposition  of  the  reasons  for  each  exercise  and  its  hygienic  and 
physiologic  function. 

LESSON  XVII 

AN  EXAMINATION  LESSON 
PART  I 

(Answer  all  five  questions) 

1.  Name  four  food  nutrients  other  than  water,  and  name  a  food  in 
which  each  nutrient  predominates. 


Divisions  of  Biology  207 

2.  Describe  the  different  kinds  of  teeth  and  their  special  adaptation 

for  their  respective  functions. 

3.  Compare  arteries  with  veins  as  to  structure  and  function. 

4.  Name  three  organs  of  excretion,  and  name  a  waste  product 

given  off  by  each. 

5.  Describe  the  effects  of  alcohol  on  the  nervous  system ;   on 

digestion. 

PART  II 

(Answer  any  three) 

6.  Why  should  we  masticate  thoroughly?    take  systematic  ex- 

ercise ?  clean  the  teeth  regularly  ?  drink  only  pure  water  ? 

7.  Compare  the  human  body  with  an  engine  in  three  particulars. 

8.  Why  must  athletes  abstain  from  the  use  of  alcohol  and  tobacco  ? 

9.  How  may  a  knowledge  of  biology  help  us  to  live  longer? 

(Touch  on  at  least  three  phases.) 

PART  III 
(Answer  any  two;  reference  to  library  or  other  books  allowed) 

10.  Discuss  some  contagious  disease  as  to  source  of  contagion, 

symptoms,  treatment,  after  effects. 

11.  By  the  use  of  the  table  on  food  values,  compute  the  food  values 

and  calories  of  a  meal  consisting  of:  one  grapefruit,  one 
boiled  egg,  two  Vienna  rolls,  one  pat  of  butter,  one  baked 
apple,  a  glass  of  milk,  and  one  doughnut. 

12.  Look  up  in  some  textbook  on  physiology  other  than  the  one 

you  use,  one  of  the  following  topics  and  report  in  detail  in 
your  own  words :  gross  structure  of  the  eye ;  gross  structure 
of  the  heart ;  gross  structure  of  a  kidney. 

An  Analysis  of  the  Suggested  Examination.  The  object  of 
such  an  examination,  divided  into  sections,  is  to  test  the  pupil 
on  his  knowledge  of  certain  facts,  on  his  power  to  answer 
thought  questions,  and  on  his  ability  to  look  up  topics  in  out- 
side reference  books  and  to  reproduce  this  knowledge  in  his  own 


208    Supervised  Study  in  Mathematics  and  Science 

words.  It  will  be  noted  that  the  questions  in  Part  I  are 
essentially  fact  questions  and  may  be  said  to  have  been  selected 
from  certain  minimum  essentials  which  should  be  required  of 
all.  The  second  part  is  composed  of  questions  which  will 
require  the  exercise  of  more  or  less  thought  and  yet  are  graded 
to  reach  pupils  of  only  average  ability.  The  third  group 
requires  special  effort  on  the  part  of  the  pupil  and  shows  his 
ability  to  use  books  other  than  the  one  he  has  been  studying. 
It  tests  his  power  of  assimilating  and  reproducing  the  author's 
material  and  of  judgment  as  to  the  selection  he  shall  make. 

It  is  expected  that  all  the  pupils  will  answer  the  questions  in 
the  first  two  groups,  but  that  only  pupils  of  more  than  average 
ability  and  resourcefulness  will  attempt  the  last  group.  It  is 
further  planned  that  the  grading  of  the  paper  be  along  these 
lines :  four  correct  answers  in  group  one  and  two  in  group  two 
are  necessary  for  a  passing  grade ;  if  all  the  answers  in  the  first 
group  and  the  three  in  the  second  group  are  correct,  the  grade 
will  be  80;  in  addition,  each  of  two  correct  answers  in  the 
third  group  will  add  ten  more  credits,  thus  giving  an  honor,  and 
if  two  are  correct  in  the  third  group,  the  result  will  be  perfec- 
tion, or  ico. 

The  advantage  claimed  for  this  kind  of  examination  papers 
is  that  it  will  allow  pupils  to  secure  a  passing  grade  through  a 
definite  mastery  of  certain  facts  of  minimum  requirement,  it 
will  give  added  value  to  the  grade  through  the  ability  of  the 
pupils  to  answer  correctly  simple  thought-producing  questions, 
and  will  enable  the  candidate  to  secure  honor  marks  only 
through  his  power  to  answer  correctly  more  advanced  questions 
after  giving  correct  answers  to  the  fact  and  thought  questions. 
It  is  held  that  the  common  form  of  examinations,  such  as  the 
Regents  Examinations  of  New  York  State,  which  allow  pupils 


Divisions  of  Biology  209 

to  range  from  60  to  100  in  their  grades,  according  to  their 
ability  to  answer  more  or  less  correctly  ten  or  more  questions 
from  a  larger  collection  —  all  of  which  are  of  about  the  same 
difficulty  and  nature  —  is  not  a  scientific  method  of  estimating 
the  pupil's  knowledge  or  judgment.  In  order  to  receive  high 
marks,  the  pupil  should  answer  questions  of  recognized  severity , 
in  addition  to  others  requiring  less  breadth  of  view  but  which 
are  more  precise  in  nature.  Such  a  plan  seems  to  encourage 
not  only  the  mastery  of  certain  minimum  requirements  but  also 
advanced  work  during  the  year,  for  only  through  such  extra 
research  and  effort  may  the  pupil  be  trained  to  secure  the  cov- 
eted high  grades.  The  old  method  seems  to  the  writer  too 
much  like  paying  the  artisan  according  to  the  number  of  times 
he  can  do  a  certain  simple  task  well  instead  of  according  to  his 
ability  to  do  more  complicated,  skilled,  and  technical  work. 
As  a  matter  of  fact  the  foreman  in  a  shoe  factory  is  not  paid  for 
his  ability  to  turn  out  a  great  number  of  heels  in  a  day,  nec- 
essary as  this  is  and  a  thing  he  is  capable  of  doing,  but  for  his 
ability  to  supervise  the  work  of  all  the  employees,  manage 
men,  and  keep  up  production.  This  higher  degree  of  skillful- 
ness  on  his  part  is  the  criterion  by  which  he  is  able  to  secure 
higher  wages ;  so  it  should  be  in  the  examination,  —  the  pupil 
receiving  the  higher  wage  or  grade  should  be  the  pupil  who  in 
addition  to  having  mastered  the  less  technical  yet  important 
knowledge  can  prove  himself  capable  of  performing  additional 
work  of  a  higher  and  more  advanced  order. 


FIFTH   SECTION 
PHYSICS 


CHAPTER  SEVEN 

FURTHER  LESSONS  IN  SCIENCE 

Space  precludes  any  extended  elaboration  of  illustrative 
lessons  in  physics,  chemistry,  physiography,  and  other  ad- 
vanced sciences.  It  is  assumed  that  the  suggestions  in  biology 
will  serve  to  illustrate  possible  lessons  in  these  subjects. 
Naturally,  as  the  pupil  advances  in  his  school  life  and  learns 
more  definitely  how  to  study,  the  need  of  extensive  directed 
study  will  be  of  less  importance  and  necessity.  For  this 
reason,  only  a  few  typical  lessons  are  given,  and  these  in 
physics.  The  illustrative  lessons  in  this  subject  may  well 
serve,  however,  as  suggestive  of  similar  work  in  the  other 
sciences. 

For  the  same  reasons,  it  is  deemed  unnecessary  to  take 
space  for  the  evaluation  of  the  content  of  these  subjects  as  has 
been  done  in  algebra,  geometry,  and  biology.  Such  an  eval- 
uation of  each  subject  into  units  of  instruction  and  units  of 
recitation  is  nevertheless  important,  and  the  teacher  will  do 
well  to  prepare  such  a  prospectus  or  syllabus  if  he  desires  to 
cover  the  work  systematically  and  with  proportionate  thor- 
oughness. 

There  is  a  concerted  effort  on  the  part  of  educational  author- 
ities to-day  toward  reorganizing  the  courses  in  all  sciences  in 
secondary  schools  along  the  line  of  the  general  needs  of  pupils 

213 


214    Supervised  Study  in  Mathematics  and  Science 

and  society l  rather  than  the  specialization  of  the  content  mat- 
ter. The  progressive  teacher  will  keep  abreast  of  these  new 
investigations  and  will  evaluate  his  courses  according  to  the 
latest  and  best  suggestions.  Any  organization  of  science 
courses  will  therefore  be  more  or  less  temporary  since  science  is 
by  nature  a  subject  of  growth  and  change.  It  is,  indeed,  this 
quality  of  progress  and  development  which  makes  the  study 
of  science  so  peculiarly  fascinating  and  vital. 

LESSON  I 

UNIT   OF   INSTRUCTION.  —  FLUIDS 

LESSON  TYPE.  —  AN  EXPOSITORY  AND  How  TO 
STUDY  LESSON 

Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 25  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Take  up  the  problems  on  pressure  of  liquids, 
assigned  for  to-day,  as  follows :  Call  on  some  pupil  to  read  the 
first  example,  to  explain  what  is  called  for,  and  to  tell  orally 
how  he  solved  it,  giving  his  answer.  If  the  answer  is  correct 
and  all  consent  to  his  solution,  pass  to  the  next  one.  If  there 
is  some  question  about  it,  however,  raised  either  by  the  teacher 
or  some  member  of  the  class,  direct  these  queries  to  the  pupil 
reciting,  making  him  substantiate  his  method  or  see  his  mistake 
if  he  is  in  error.  Avoid  telling  him  where  he  made  his  mistake 
as  he  will  never  gain  the  power  of  problem  solving  unless  he  is 
directed  in  the  finding  of  his  own  error  and  also  directed  how 

1  "  Reorganization  of  science  in  secondary  schools,"  Bulletin  No.  26,  1920, 
Federal  Bureau  of  Education. 


Further  Lessons  in  Science  215 

to  correct  the  same.  Each  problem  explained  should  be  the 
means  of  clarifying  and  forcibly  demonstrating  the  principles 
involved.  Explaining  them  singly  and  orally  concentrates 
the  attention  of  the  entire  class  and  brings  out  any  wrong 
impressions  or  false  methods  of  solution  that  may  have  been 
practiced  by  anyone.  It  will  also  serve  to  clarify  in  the  pupil's 
mind  the  process  of  reasoning  which  must  mark  all  attempts  at 
solving  problems  in  physics. 

If  the  time  for  the  review  is  limited  as  in  the  time  schedule 
noted  above,  it  will  expedite  matters  to  take  up  only  those 
exercises  which  gave  trouble.  As  a  general  thing  this  is  the 
best  way  at  all  times,  for  there  is  little  value  in  explaining 
problems  which  all  have  successfully  solved,  unless  it  be  to 
make  sure  that  they  all  solved  them  by  correct  methods. 

The  Assignment.  —  Method.  Put  a  fresh  egg  in  some  fresh 
water,  so  that  the  class  may  clearly  see  that  the  egg  sinks. 
Then  put  it  in  a  glass  of  a  saturated  saline  solution.  Ask  the 
class  why  it  does  not  sink  now.  In  the  same  manner  put  a 
marble  in  the  glass  of  fresh  water  and  also  in  some  mercury. 
Ask  someone  if  in  swimming  he  has  ever  noticed  how  he  will 
with  difficulty  keep  his  feet  if  he  wades  out  slowly  into  the 
water  up  to  his  mouth.  Ask  someone  who  may  have  lived  or 
been  near  the  ocean,  if  he  experienced  the  same  effect  in  salt 
water.  These  simple  experiments  will  serve  to  emphasize  the 
fact  that  bodies  submerged  in  a  liquid  will  be  forced  upward, 
the  degree  depending  upon  the  nature  of  the  liquid  and  the 
nature  of  the  body  submerged.  Ask  someone  to  state  this 
fact  in  the  form  of  a  simple  statement  or  rule.  If  the  statement 
as  given  is  not  complete,  draw  out  the  nature  of  its  incomplete- 
ness by  skillful  questioning.  In  other  words,  teach  the  pupils 
how  to  gather  the  results  of  experimentation  and  study  and 


216    Supervised  Study  in  Mathematics  and  Science 

make  deductions  therefrom.  Then  when  they  read  in  their 
textbook  similar  conclusions  they  will  feel  a  thrill  of  power  in 
their  ability  to  conclude  results  for  themselves.  This  idea 
should  be  the  predominating  object  of  the  explanation  of  the 
assignment  —  to  develop  through  the  pupils'  own  observation 
and  research  the  sensing  of  the  laws  and  facts  of  physics. 

Ask  if  anyone  knows  who  Archimedes  was.  If  none  knows 
anything  about  this  great  scientist,  direct  someone  to  secure 
the  story  of  his  life  in  an  encyclopedia.  While  he  is  doing 
this,  explain  the  meaning  of  the  word  buoyancy,  using  as  an 
illustration  the  word  buoy,  which  all  will  probably  understand. 
Ask  for  illustrations  of  the  application  of  buoyant  force,  which 
might  well  include  the  floating  of  logs  down  a  river,  the  float- 
ing of  a  boat,  the  fact  that  a  drowning  person  will  come  to  the 
surface  two  or  three  times,  etc.  Referring  to  the  report  of 
the  pupil  concerning  Archimedes  and  the  story  of  his  life  ex- 
plain that  he  was  the  first  person  to  discover  certain  laws 
which  govern  buoyancy.  Mention  the  fact  that  we  shall 
try  to  discover  these  laws  for  ourselves. 

Weigh  some  metal,  as  a  piece  of  lead,  in  air  and  then  weigh 
it  when  submerged  in  water.  Call  on  some  pupil  to  come  for- 
ward and  announce  aloud  the  readings  of  the  scales  in  each 
case.  Ask  what  we  understand  by  weight.  If,  then,  it  is  the 
force  of  attraction  between  the  body  and  the  earth,  ask  if  the 
piece  of  metal  has  really  lost  any  weight.  The  pupils  will 
readily  see  that  this  is  impossible  according  to  the  definition. 
Ask  them  what  has  caused  the  apparent  loss  of  weight.  The 
class  will  be  quick  to  see  that  there  has  been  some  readjustment 
of  the  weight  rather  than  any  loss.  Now  perform  the  same 
experiment,  using  in  this  case  a  cylindrical  dish  which  will 
hold  exactly  a  solid  of  similar  form.  Then  if  we  allow  the 


Further  Lessons  in  Science  217 

weight  to  be  submerged  in  water  and  fill  the  receptacle 
with  the  displaced  water,  the  equilibrium  will  be  restored.  It 
will  be  a  slow  class  that  will  not  at  once  see  that  the  weight 
lost  in  water  an  amount  equal  to  the  weight  of  the  water  it 
displaced.  Now  have  someone  give  this  in  terms  of  a  state- 
ment. After  two  or  three  have  done  this,  each  time  making 
the  statement  more  complete  and  clear,  tell  them  they  have 
evolved  one  of  the  laws  of  Archimedes. 

Referring  to  the  previous  experiment  of  putting  the  egg  in 
the  brine,  ask  them  what  it  did  in  the  solution.  They  will  tell 
you  it  floated.  Ask  them  what  caused  the  egg  to  sink  in  the 
fresh  water  but  not  in  the  brine.  Perform  the  experiment, 
similar  to  the  one  above,  for  sinking  bodies.  It  is  clearly  given 
in  all  textbooks  and  need  not  be  stated  here.  The  principle 
for  floating  bodies  may  also  be  deduced  as  was  done  with  the 
above  principle  of  buoyancy. 

Ask  someone  to  tell  what  is  meant  by  mass.  Stating  that 
the  quantity  of  matter  or  the  mass  in  a  unit  volume  measures 
the  density  or  comparative  density  of  a  solid,  ask  if  we  could 
tell  the  density  of  a  piece  of  iron,  for  instance,  by  weighing  it. 
Presumably  not,  for  it  is  possibly  irregular  in  character  and  its 
weight,  depending  on  its  size,  will  vary,  while  its  density  would 
always  be  the  same.  Therefore,  as  in  all  measurements,  we 
must  take  some  unit  and  we  shall  find  that  the  density  may  be 
found  by  dividing  the  mass  by  the  volume.  Put  this  on  the 
board,  first  as  represented  by  words,  as 

j      .,          mass 
density  =  — , 

volume 
and  then  as  a  formula : 

,    m 
a  =  — . 


2 1 8    Supervised  Study  in  Mathematics  and  Science 

Weigh  a  piece  of  iron  which  may  be  measured  and  the  volume 
of  which  may  be  computed  in  c.c.,  and  then  divide  the  one  by 
the  other.  When  this  is  done,  ask  someone  to  turn  to  the 
table  of  density  or  specific  gravity  in  his  book  to  compare  the 
result  with  that  given.  They  will  be  very  much  gratified  to 
find  their  result  tallies  with  that  given  in  the  book.  Let  some 
other  pupils  repeat  the  operation  with  another  piece  of  iron; 
then  with  some  lead,  marble,  etc. 

Noting  that  both  of  these  substances  are  regular  in  surface, 
ask  how  they  would  suggest  proceeding  with  some  irregular  sub- 
stance, as  a  piece  of  quartz.  Someone  may  be  quick  enough 
to  suggest  that  we  might  get  its  weight  by  the  method  of  water 
displacement,  and  then  substitute  in  the  formula  as  before. 

Again  note  that  these  substances  are  heavier  than  water, 
and  ask  someone  to  suggest  a  method  for  finding  the  specific 
gravity  of  something  lighter  than  water,  as  paraffin.  Again 
someone  will  probably  suggest  that  we  attach  a  known  weight 
to  the  paraffin  and  submerge  both  in  water,  then  subtract  the 
known  weight  from  the  gross  weight  and  proceed  as  before. 
Thus  we  have  outlined  the  methods  of  ascertaining  the  specific 
gravity  of  solids. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 
The  textbook  work  on  the  subject. 

//  or  Average  Assignment.  The  questions  found  at  the  end 
of  the  chapter  covering  this  subject. 

Ill  or  Maximum  Assignment. 

1.  Tabulate  in  order  of  their  density :   tin,  ice,  gold,  zinc, 
glass,  and  butter. 

References :   any  chemistry. 

2.  What  is  the  principle  of  the  submarine? 

3.  Look  up  about  Descartes  and  his  Cartesian  diver. 


Further  Lessons  in  Science  219 

LESSON  II 

UNIT   OF  INSTRUCTION.  —  FLUIDS 

LESSON  TYPE.  —  A  LABORATORY  LESSON 
Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 25  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Conduct  a  short,  snappy  recitation  on  the 
textbook  work  on  density  and  specific  gravity  of  solids.  This 
review  should  serve  as  a  preliminary  review  and  preparation 
for  the  individual  experiment. 

The  Assignment.  —  During  the  assignment  the  instructor 
should  outline  how  the  experiment  is  to  be  made  and  what 
computations  and  deductions  must  enter  into  the  written  record. 

Method.  The  instructor  should  write  on  the  blackboard  with 
yellow  crayon  the  object  of  the  experiment  as  follows : 

Object.  To  compare  the  buoyant  effect  on  a  solid  submerged 
in  the  water  with  the  weight  of  the  water  which  is  displaced. 

Under  the  caption  Apparatus  for  the  present  no  data  are  to 
be  placed.  It  is  best  to  leave  this  to  be  filled  after  the  actual 
experiment  has  been  made,  so  that  the  pupil  may  sense  the 
real  object  of  these  data,  which  is  to  be  the  recording  of  the 
apparatus  used  rather  than  some  arbitrary  listing  of  equip- 
ment. It  is  as  premature  to  list  the  articles  of  apparatus  be- 
fore the  experiment  is  made  as  it  would  be  to  attempt  to  count 
the  votes  before  an  election.  How  many  mechanics  could  tell 
prior  to  doing  some  repair  work  on  an  automobile  just  what 
tools  were  going  to  be  used? 
Under  Procedure,  explain  to  the  class  in  brief  outline  how 


22o    Supervised  Study  in  Mathematics  and  Science 

the  actual  experiment  is  going  to  be  made,  but  by  no  means 
dictate  any  directions  for  its  performance.  This  must  be 
written  up  in  the  language  of  the  pupils  and  is  to  follow  the 
experiment.  Explain  that  the  method  employed  will  be  to 
weigh  the  solid  in  air  and  then  in  water,  and  that  the  best 
method  of  doing  this  will  be  to  allow  the  scales  to  project  over 
the  table  a  little  so  that  the  solid  may  be  attached  to  the  under 
side  of  one  scale  and  weighed.  Then  we  shall  take  an  over- 
flow can  and  fill  it  with  water,  holding  the  finger  over  the 
spout.  Now  after  allowing  all  superfluous  water  to  run  out, 
we  shall  place  it  under  the  projecting  solid  and  then  allow  the 
solid  to  drop  into  the  bucket,  catching  the  overflow  of  water  in  a 
catch  bucket.  When  the  body  is  entirely  submerged,  weigh  the 
solid  in  this  position.  Explain  that  care  must  be  taken  to  catch 
every  drop  of  water  and  also  to  be  as  accurate  as  possible  in 
every  weighing  operation.  Also  suggest  that  other  methods 
of  arranging  the  scales  for  this  experiment  may  be  made  as  the 
pupils  desire,  such  as  supporting  the  scales  on  a  pile  of  books, 
a  box,  etc. 

Then  ask  some  pupil  to  state  what  data  will  have  been  col- 
lected up  to  this  point.  Step  to  the  board  and,  as  these  are 
stated,  write  them  down.  They  will  be : 

1.  Weight  of  the  solid  in  air. 

2.  Weight  of  the  solid  submerged  in  water. 

3.  Weight  of  the  empty  catch  bucket. 

4.  Weight  of  the  catch  bucket  and  the  displaced  water. 

If  the  four  mentioned  above  are  given  in  different  order, 
write  them  down  as  given  and  afterward  ask  whether  any  would 
change  the  order,  according  to  logical  steps.  The  above  order 
will  probably  be  suggested.  It  will  be  noted  by  the  reader 
that  at  every  step  the  pupil  is  thrown  on  his  own  judgment  as 


Further  Lessons  in  Science  221 

to  the  order  of  procedure  and  not  merely  told  these  steps  arbi- 
trarily. He  will  thus  be  trained  to  exercise  his  own  resources 
and  judgment. 

When  these  tabulations  have  been  arranged  for  satisfactor- 
ily, ask  some  pupil  to  tell  you  what  computations  must  be 
made  in  order  to  draw  a  conclusion.  With  a  little  skillful 
leading  he  mil  see  that  we  must  compute  from  the  figures 
found :  (a)  the  loss  of  weight  of  the  solid  in  water,  and  (&)  the 
weight  of  the  water  displaced  by  the  submerged  solid.  Using 
some  fictitious  numbers  for  data,  have  him  tell  you  these  two 
computations. 

Suppose  for  illustration,  you  assign  these  amounts  to  his 
list  of  data : 

1.  Weight  in  air 15  gm. 

2.  Weight  in  water 10  gm. 

3.  Weight  of  bucket 5  gm. 

4.  Weight  of  bucket  with  the  displaced  water       .     .  10  gm. 

Then  our  computations  will  be : 

1.  Loss  of  weight  of  solid  in  water :  15  gm.  — 10  gm.  =  5  gm. 

2.  Weight  of  water  displaced :   10  gm.  —  5  gm.  =  5  gm. 

What  then  may  we  conclude  from  these  computations? 
How  do  the  two  results  compare?  The  pupils  will  of  course 
see  that  they  are  the  same. 

Ask  someone  to  express  this  conclusion  in  the  form  of  a 
statement.  How  does  this  agree  with  Archimedes'  principle  ? 

Now  ask  someone  to  state  the  rule  or  formula  for  density. 

It  is 

,    m 
d=T 

or  density  equals  the  mass  divided  by  the  loss  of  weight  in 
water. 


222    Supervised  Study  in  Mathematics  and  Science 

Using  the  data  of  the  above,  have  some  pupil  compute  the 
density  of  the  solid,  thus,  15-^-5=3. 

The  Study  of  the  Assignment.  —  Now  let  each  pupil  get  the 
apparatus  he  needs,  make  the  experiment  with  various  solids, 
and  write  up  his  experiment  in  the  form  suggested.  Insist 
that  each  pupil  get  his  own  apparatus,  and  clean  it  up  and  put 
it  away  after  the  work  is  completed.  The  author  has  no 
sympathy  with  the  method  by  which  some  teachers  conduct  a 
laboratory  experiment,  where  all  of  the  apparatus  is  set  out 
before  the  class  and  blanks  are  distributed  to  the  pupils,  who 
participate  only  in  the  experiment  by  inserting  the  data  in  the 
blank  spaces.  This  method  is  too  much  along  the  line  of 
"  press  the  button  and  we  do  the  rest  "  photography.  Pupils 
taught  in  this  manner  may  perform  experiments  until  dooms- 
day and  they  will  know  no  more  physics  at  the  end  than  they 
did  at  the  beginning.  The  main  value  of  the  individual  labora- 
tory experiment  is  in  the  fact  that  the  actual  procedure  makes 
a  more  lasting  impression  on  the  pupil  than  the  mere  reading 
of  the  experiment  out  of  some  book  or  witnessing  a  demonstra- 
ation  made  by  the  instructor. 

Drawings.  When  drawings  may  further  explain  just  how 
the  experiment  was  done,  they  are  important,  but  when  the 
process  is  self-evident,  or  easily  explained,  drawings  are  a  waste 
of  time.  When  required  they  should  be  done  freehand,  with 
the  fewest  lines  possible  and  they  should  be  clearly  explained 
by  the  accompanying  legend.  Copied  drawings  are  valueless 
as  they  do  not  function  as  they  ought ;  the  pupil  is  thinking 
more  of  the  technic  of  the  figure  than  of  the  object  for  which  it 
is  drawn ;  namely,  graphically  to  illustrate  and  supplement  the 
written  description. 

During  the  performance  of  the  experiment  by  the  pupil,  the 


Further  Lessons  in  Science  223 

teacher  should  pass  about  to  see  that  everything  is  being 
done  correctly,  ready  with  suggestions  but  never  actually 
telling  the  pupil  how  he  may  better  his  work.  In  other  words, 
the  work  of  the  laboratory  should  be  used  to  develop  the  pupil's 
ability  to  understand  the  textbook,  to  appreciate  the  value  of 
careful  manipulation  of  apparatus,  and  to  make  possible  the 
logical  drawing  of  conclusions  as  a  result  of  the  work  per- 
formed. 

LESSON  III 

UNIT   OF  INSTRUCTION.  —  FLUIDS 

LESSON  TYPE.  —  A  How  TO  STUDY  LESSON 
IN  PROBLEMS 

Program  or  Time  Schedule 

The  Review 10  minutes 

The  Assignment 25  minutes 

The  Study  of  the  Assignment 25  minutes 

The  Review.  —  Call  on  some  pupil  to  step  to  the  front  of  the 
room  and  briefly  review  his  entire  work  in  the  laboratory 
yesterday  without  referring  to  his  notebook  except  for  figures. 

The  Assignment.  —  The  best  way  to  take  up  a  set  of  new 
problems  in  physics  is  to  have  the  class  orally  analyze  a  number 
of  simple  exercises,  such  as  will  be  found  in  every  textbook, 
thus  giving  a  large  number  of  pupils  practice  in  interpreting 
what  is  wanted.  It  is  also  well  to  have  a  number  of  supple- 
mentary problems  from  other  sources,  which  may  be  used  in 
this  way,  if  the  instructor  finds  that  the  pupils  do  not  readily 
grasp  the  principles  involved.  A  large  assortment  of  applied 
problems  may  be  found  in  Lynde's  Physics  of  the  Household.1 
1  The  Macmillan  Company,  1918. 


224    Supervised  Study  in  Mathematics  and  Science 

It  is  found  that  the  greatest  difficulty  pupils  have  with 
problems  is  their  inability  to  interpret  correctly  the  language 
of  the  particular  problem ;  the  actual  mathematics  is  usually 
very  simple.  It  is  for  this  reason  that  the  oral  analysis  is  an 
exceedingly  good  method  of  training  pupils  to  make  the  proper 
interpretation. 

It  will  be  found  expedient  to  have  some  pupil  read  a  certain 
problem  aloud;  then  after  giving  him  a  few  seconds  for 
thought,  ask  him  to  shut  his  book  and  repeat  the  essential 
points  of  the  problem.  If  he  can  state  the  problem  in  his  own 
words,  the  instructor  may  feel  assured  that  he  understands 
what  it  is  about,  unless  it  be  memorized.  Now  have  him 
state  what  is  to  be  found.  With  each  step  of  his  solution, 
insist  on  his  giving  a  physical  reason  for  the  step.  The 
teacher  should  never  accept  such  bald  statements  as  "  mul- 
tiply so  and  so  and  divide  the  result  by  —  ,"  unless  the 
pupil  gives  as  his  reason  for  so  doing  some  rule  or  physical 
fact.  Also,  the  answer  should  be  given  in  some  denomina- 
tion, so  that  the  instructor  may  know  that  the  pupil  has 
understood  the  purpose  of  the  problem.  Thus  the  analyses  of 
the  exercises  become  an  excellent  review  of  the  principles  of 
physics. 

The  exercises  which  are  assigned  for  the  next  lesson  may  be 
some  of  those  taken  up  thus  analytically  in  class,  as  well  as 
some  others  of  similar  and  possibly  more  severe  character. 
These  should  be  worked  out  on  paper  neatly  and  with  logical 
steps,  omitting  of  course  much  of  the  reasoning  that  has  been 
done  in  the  oral  work. 

The  Study  of  the  Assignment.  —  I  or  Minimum  Assignment. 
The  first  ten  of  the  fifteen  problems  in  the  textbook. 

//  or  Average  Assignment.     The  remaining  five  problems. 


Further  Lessons  in  Science  225 

777  or  Maximum  Assignment.  Problems  18-22,  page  48, 
Lynde's  Physics  of  the  Household.1 

The  Silent  Study.  —  During  this  part  of  the  period  the 
instructor  will  find  that  he  will  be  needed  by  some  of  the 
weaker  pupils  who  have  still  failed  to  master  the  solution  of  the 
problems,  either  through  a  faulty  understanding  of  the  laws 
or  of  the  principles  involved.  The  teacher  will  find  this  an 
opportunity  for  individual  help  and  as  has  been  emphasized 
many  times  throughout  these  lessons,  he  must  ever  be  on  the 
alert  not  to  tell  how  to  do  them  but  to  lead  the  pupil  to  solve 
them  himself. 

All  exercises  should  be  collected  at  the  close  of  the  hour,  with 
notice  that  the  remainder  will  be  collected  to-morrow.  Thus 
the  teacher  will  know  that  the  ones  handed  in  are  indeed  the 
work  of  the  pupils  themselves,  and  by  examining  them  he  may 
get  a  clear  idea  of  just  how  proficient  they  are  becoming  in 
handling  this  work.  The  exercises  done  outside  of  class  are 
of  doubtful  value,  except  as  they  reflect  the  pupil's  mastery  of 
the  principles  as  he  later  shows  by  his  ability  to  do  others  of 
similar  nature.  They  must  be  required,  however,  in  schools 
which  do  not  have  periods  lasting  more  than  sixty  minutes, 
and  they  should  be  carefully  checked  and  as  far  as  possible 
carefully  examined.  They  are  of  doubtful  value  in  awarding 
grades,  but  the  failure  to  hand  them  in  or  incorrect  solutions 
should  be  reflected  in  the  grades  awarded. 

LESSON  IV 

RED   LETTER  DAY  LESSONS 

From  time  to  time  throughout  the  course,  there  should  be 
provided  special  programs  or  red  letter  day  lessons.  Instead 

1  The  Macmillan  Company. 


226     Supervised  Study  in  Mathematics  and  Science 

of  limiting  this  lesson  to  any  one  program,  a  number  of  sug- 
gestive lessons  will  be  mentioned,  with  a  few  words  of  explana- 
tion concerning  each.  The  progressive  teacher  of  physics 
will  think  of  many  others,  of  course,  and  he  should  choose  the 
kinds  that  seem  to  be  of  most  value  and  interest  to  his  partic- 
ular class. 

1.  A  stereopticon  lecture  on  some  phase  of  physics.     Suit- 
able slides  are  sold  by  a  number  of  firms,  as  L.  E.  Knott 
Apparatus  Co.,  Boston;   Central  Scientific  Co.,  Chicago. 

2.  If  the  school  owns  a  moving  picture  machine,  many 
valuable  films  may  be  secured   from   various  sources.     See 
Extension  Leaflet  No.  2,     Department  of  the  Interior,  for 
list  (December,  1919). 

3.  Cuts  from  magazines  and  books,  photos,  etc.  may  be  used 
to  advantage  in  giving  a  review  of  some  unit  of  instruction 
through  the  use  of  an  opaque  projector. 

4.  A  talk  to  the  class  by  some  college  professor  of  physics ; 
by  the  city  electrician  on  some  practical  phase  of  his  work; 
by  some  physician  on  the  X-ray  in  surgery ;  by  a  musician  on 
the  pipe  organ ;  etc. 

5.  A  trip  to  the  electric  power  plant,  or  to  the  pump- 
ing  station,    or    to    a    plant    using    a    hydrostatic    press, 
etc. 

6.  A  wireless  apparatus  may  be  set  up  in  the  laboratory  and 
the  hour  spent  in  sending  and  receiving  messages. 

7.  An  examination  of  various  kinds  of  vacuum  cleaners, 
either  expository  by  the  teacher  or  the  actual  examination  of 
samples  which  may  be  collected  from  different  sources  and 
loaned  for  the  day.     Excellent  cuts  may  be  found  in  Lynde's 
Physics  of  the  Household. 1 

1  The  Macmillan  Company. 


Further  Lessons  in  Science  227 

8.  Some  pupil  may  give  an  exposition  of  some  article  on 
an  appropriate  subject  taken  from  a  current  magazine,  as 
Scientific  American  or  Popular  Science  Monthly. 


BIBLIOGRAPHY 

I.    BOOKS  FOR  THE  TEACHER  OF  MATHEMATICS 
AND  SCIENCE 

Belts,  G.  H.  —  "The  Recitation";  Houghton  Mifflin  Co.,  1911. 
Cajori,  Florian  —  "A    History    of    Mathematics";     The    Macmillan 

Company,  1919. 
Earhart,  Lida  B.  —  "Teaching  Children  to  Study"  ;  Houghton  Mifflin 

Co.,  1909. 

Earhart,  Lida  B.  —  "Types  of  Teaching"  ;  Houghton  Mifflin  Co.,  1915. 
Evans,  G.  W.  —  "The  Teaching  of  High  School  Mathematics"  ;  Hough- 

ton  Mifflin  Co.,  1911. 
Hall-Quest,  A.  L.  —  "Supervised  Study";  The  Macmillan  Company, 

1916. 
Johnston  and  Others  —  "High  School  Education";    Chas.  Scribner's 

Sons,  1912. 
Judd,  C.  H.  —  "Psychology  of  High  School  Subjects";  Ginn  and  Co., 


Lloyd   and  Bigelow  —  "The  Teaching  of  Biology  in  the  Secondary 

School";   Longmans,  Green  and  Co.,  1904. 
Mann,  C.  R.  —  "The  Teaching  of  Physics"  ;  The  Macmillan  Company, 

1912. 
McMurry,  F.  M.  —  "How  to  Study  and  Teaching  How  to  Study"; 

Houghton  Mifflin  Co.,  1909. 

Milner,  Florence  —  "The  Teacher";  Scott,  Foresman  and  Co.,  1912. 
Parker,  S.  C.  —  "Methods  of  Teaching  in  High  Schools"  ;  Ginn  and  Co., 


Sanford,  Fernando  —  "How  to  Study,  Illustrated  through  Physics"; 

The  Macmillan  Company,  1922. 
Schultze,    Arthur  —  "The   Teaching   of    Mathematics    in    Secondary 

Schools";  The  Macmillan  Company,  1912. 
Smith  and  Hall  —  "  The  Teaching  of  Chemistry  and  Physics  in  Second- 

ary Schools  "  ;  Longmans,  Green  and  Co.,  1904. 

229 


230    Supervised  Study  in  Mathematics  and  Science 

Smith,  D.  E.  —  "The  Teaching  of  Elementary  Mathematics";  The 
Macmillan  Company,  1917. 

Smith,  D.  E.  —  "The  Teaching  of  Geometry";   Ginn  and  Co.,  1911. 

Stoner,  W.  S.  —  "Natural  Education" ;  Bobbs-Merrill  Co.,  1914. 

Strayer,  G.  D.  —  "Brief  Course  in  the  Teaching  Process";  The  Mac- 
millan Company,  1912. 

Young,  J.  W.  A.  —  "The  Teaching  of  Mathematics  in  the  Ele- 
mentary and  Secondary  School"  Longmans,  Green  and  Co.,  1911. 

II.    MAGAZINES  FOR  MATHEMATICS  AND  SCIENCE 
TEACHERS 

The  Mathematics  Teacher,  41  North  Queen  St.,  Lancaster,  Pa. 

School  Science  and  Mathematics,  2059  E.  72d  St.,  Chicago,  111. 

Science,  The  Science  Press,  Garrison,  N.  Y. 

General  Science  Quarterly,  Salem,  Mass. 

School  Review,  University  of  Chicago  Press,  Chicago,  HI. 

Bulletins  of  the  United  States  Bureau  of  Education,  Washington,  D.  C. 

No.  3.     Science  Teaching  in  the  Secondary  Schools. 

No.  4.     Mathematics  in  the  Secondary  Schools. 

No.  8.     Examinations  in  Mathematics. 

No.  12.   Training  Teachers  of  Mathematics. 

No.  14.  Report  of  the  American  Commission  on  Teaching  of  Mathe- 
matics. 

No.  16.   Mathematics  in  Public  and  Private  Schools. 

No.  26.   Reorganization  of  Science  in  Secondary  Schools. 
Monthly  Record  of  Current  Educational  Publications. 

III.    STANDARD  TESTS  AND  MEASUREMENTS 
ALGEBRA : 

Hotz's   Algebra   Scales,     First  Year;     Teachers  College,    Columbia 

University,  New  York  City. 
Rugg  and  Clark's  Standardized  Tests  in  First  Year  Algebra;  University 

of  Chicago  Press,  Chicago,  111. 
Thorndike's  Algebra  Test;    Teachers  College,  Columbia  University, 

New  York  City. 


Bibliography  231 

GEOMETRY : 

Minnick's  Geometry  Tests;  University  of  Pennsylvania,  Philadelphia, 

Pa. 

Rogers'  Mathematical  Tests;  Teachers  College,  Columbia  University, 
New  York   City. 

PHYSICS  : 
Starch's  Tests  in  Physics;  University  of  Wisconsin,  Madison,  Wis. 


INDEX 


Absences,  elimination  of,  40-41 

Accuracy,  18,  65,  109,  122 

Activities,  muscular,  203,  204 

Adaptations,  158,  177 

Adding  machine,  30 

Addition,  associative  law  of,  61 ; 
commutative  law  of,  60-61 

Agriculture,  33 

Air,  fresh,  44 

Airplane,  30,  172 

Algebra,  48,  213;  applications  of,  36; 
bird's-eye  view  of  course  in,  33; 
Comte's  definition  of,  35 ;  divi- 
sions of,  20-24  5  function  of,  36,  54 ; 
history  of,  27-28;  interrelationship 
of  arithmetic  and,  34;  methods  of 
teaching  intermediate,  141-145 ; 
methods  of  teaching  advanced, 
141-145 ;  necessity  of,  31-32 ;  ori- 
gin of  the  word,  27 ;  practical  value 
of,  29-32;  problems  as  real  tests 
in,  96-97;  quotation  from  Milne's 
Standard,  54;  representation  of 
things  concrete  in,  75-78;  Sir 
Isaac  Newton's  definition  of,  35; 
solving  problems  in,  46;  speed 
tests  in,  73-75;  "spelling-down 
bee"  in,  80-81;  standardized  tests 
foj  74-75 !  technic  of  textbook  in, 
47-48 ;  textbook  in,  51 ;  time 
table  for,  25 

Al-jebr  w'al  muqubalah,  27 

Allen,  L.  M.,  10 

Amortization  of  interest-bearing  notes, 
3° 

Angles,  properties  of,   iio-ni,   123; 


questions  on,  123;  theorem  for 
vertical,  112,  117 

Animals,  snapshots  of,  196 

Answers,   use   of,   48;    complete,   55 

Ants,  182 

Aphids,  182 

Apparatus,  construction  of,  152 ;  physi- 
cal, 219-223;  special,  151;  tinker- 
ing with,  150;  use  of  gymnastic, 
204;  wireless,  226 

Aquarium,  171 

Arches,  108 

Archimedes,  216-217,  221 

Architecture,  31,  137 

Arithmetic,  common  errors  in,  36; 
Comte's  definition  of,  35 ;  examples 
i*1*  37>  38-39;  formulas  used  in, 
36;  interdependence  of  algebra 
and,  37-38;  nomenclature  of,  37; 
pupil's  present  knowledge  of,  60-6 1, 
77 ;  review  of  fundamental  processes 
in,  36-37 

Articles,  magazine  and  newspaper, 
34,  160,  186,  226,  227 

Assignment,  6  (see  also  each  lesson 
outlined);  aim  of,  12;  average, 
15,  64,  67,  132,  151 ;  completion 
of,  12-13,  64;  explanation  of  the 
new,  133;  importance  of,  12; 
maximum,  15,  64,  67,  68,  117-118, 
122, 132, 151, 199;  minimum,  14-15, 
64,  67,  68,  129,  132,  151 ;  nature  of, 
12;  study  of  the  maximum,  121- 
122;  study  of,  12,  133,  162-165, 
167-168,  I73-I7S,  177-178, 183-184, 

I9I-I92,      202,      215-218,      222-223; 


233 


234 


Index 


summary  of,  41 ;  summary  on  the 
study  of,  42;  the  threefold,  14-15, 
151;  time  allotted  to,  12 

Assignment  sheet,  how  to  make,  14, 
63-64,  72,  151 ;  how  to  use,  15-16, 
128,  130;  illustration  of,  19,  62; 
object  of,  13-14;  the  threefold,  14 

Astronomy,  31 

Attention,  individual,  69 

Authorities,  varied  opinions  of,  155 

Automobile,  30,  219 

Axioms,  105 

Bacteria,  156 

Banking,  8,  182-183,  186,  191 

Bibliographies,  186 

Biennial,  171 

Biology,  animal,  156;  conducting  a 
field  trip  in,  179-181;  correlation 
of  English  and,  205;  divisions  of, 
155-156;  equipment  of  classroom 
in,  177-178,  196,  201;  human,  156; 
lessons  in,  213;  plant,  155-156; 
problems  of,  159,  166-167 ;  survey 
of  the  course  in,  159-160;  use  of 
notebooks  in,  162 ;  valuable  lessons 
of,  157-160 

Bird  houses,  196-197 

Birds,  156,  1 60;  stereopticon  lecture 
on,  196;  stories  about,  187;  study 
of,  178 

Blackboard,  use  of,  16,  27,  39,  40,  41, 
42,  47,  56-57,  60,  63,  69,  71,  72,  77, 
79,  80,  82,  84,  87,  92,  94-95,  "I, 
113,  117,  119,  122,  123-125,  129- 
130,  134-135,  152,  158,  164,  169, 
171,  184,  187,  190-191,  193,  196, 
200-202,  202-203,  219-220;  use  of 
the  spherical,  145 

Blood,  156 

Bonds,  valuation  of  debenture,  30 

Bones,  156,  197 

Book,  the  open,  47-48 

Books,  160,  186,  193,  226;  supple- 
mentary, 1 86 


Bordeaux  mixture,  192 
Botany,  155,  195 
Bridges,  108 
Bulletins,  186 
Buoyancy,  216-217 
Burbank,  Luther,  159 
Business,  statistics  of,  79 
Busy  work,  15 
Butterfly,  182 

Cage,  157,  177,  183 

Calories,  207 

Camera,  hunting  with,  197 

Cancellation,  37 

Canton,  N.  Y.,  10,  13,  68 

Capitol,  at  Washington,  108 

Carbon,  163,  164 

Cards,  index,  143-144;  problems 
written  on,  191 ;  reviewing  Book  I 
through  use  of,  134-135;  use  of 
the  divided,  114-116,  119-122 

Carelessness,  14 

Carpenter,  58 

Catcher,  86,  87 

Caterpillar,  tent,  193 

Ceilings,  steel,  108 

Charts,  196;  making,  151,  171,  173, 
174,  175,  182,  184,  187,  195 

Checking,  value  of,  66,  245 

Chemistry,  31,  203 

Children,  educating  all  the,  151 

Circle,  105-106,  109,  145;  circum- 
ference of,  38 

Circulation,  156 

Class,  testing  progress  of,  75 

Cochineal  bug,  192 

Codling  moth,  186,  192,  193 

Coefficient,  61 

Coleoptera,  order  of,  190 

Coloration,  protective,  158 

Comte,  35 

Concentration,  44-45,  74,  108 

Cone,  109 

Contents,  table  of,  7,  47 

Contests,  33 


Index 


235 


Contractor,  32 

Corn,  kernel  of,  171,  173 

Cotton-boll  weevil,  192,  193 

Course,  7 ;  bird's-eye  view  of,  26,  33, 
109 

Course  of  study,  evaluation  of,  20, 
IO5>  IS5>  2I3>  minimum  essentials, 
208 

Court,  the  class  as  a,  89-91 

Crayfish,  182,  195 

Crayon,  use  of  colored,  84,  190,  196, 
201,  219;  waste  of,  41;  yellow,  71 

Credit,  awarding  extra,  64-65,  101 

Crustaceans,  156,  160,  181-182;  ques- 
tions on,  182 

Current  events,  33 

Curriculum,  6;  college  preparatory, 
7;  domestic  science,  7 

Cylinder,  109 

Dandelions,  181 

Darwin,  Charles,  182 

Definitions,  21,  51,  105 

Density,  216-218;  formula  for,  217 

Descartes,  218 

Devices,  6,  9,  43,  71 

Diamonds,  164 

Dictionary,  use  of,  170 

Dietitian,  31 

Digestion,  organs  of,  156 

Digits,  35 

Diophantus,  28 

Diptera,  order  of,  190 

Discipline,  180;    formal,  108 

Dividends,  distribution  of,  30 

Doughnuts,  88,  207 

Drawings,  109,  136,  177,  188,  195,  201, 
222 

Drill,  82;  function  of,  42;  impor- 
tance of,  71 

Earhart,  Lida,  8,  46 
Earth,  size  of,  107 
Education,  29,  48 


Egyptians,  28,  107,  no 

Elections,  33,  79 

Embryo,  172-173,  176 

Encyclopedia,  31 

Endosperm,  meaning  of,  172 

Engineering,  30,  31,  107 

English,  technical,  205;  use  of  pure, 
205 

Enthusiasm,  arousing,  26 

Entomologist,  class,  193 

Environment,  correct,  44;  definition 
of,  170;  importance  of,  158;  of  the 
pupil,  178,  186;  varieties  of,  171 

Equation,  73-75 ;  applications  of, 
77;  cubic,  29;  quadratic,  20,  24, 
55,  75,  92-93;  simple,  20,  23,  38; 
study  of,  75-77 

Equipment,  6,  151,  196,  201 

Euclid,  107,  109,  no 

Evolution,  20,  23 

Examination,  a  sample,  98-100,  206- 
207;  an  analysis  of  the  suggested, 
207-209;  value  of  the  suggested, 
loo-ioi ;  criticism  of  the  ordi- 
nary, 208;  final,  15;  formal,  4; 
grading,  74;  object  of,  97;  pre- 
academic,  37;  regents,  3,  208-209; 
standardized  tests  as,  97-98; 
written,  9,  97-98,  166 

Excursions,  field,  162,  194;  how  to 
conduct,  179-181;  importance  of, 
179 

Exercises,  oral,  55;  treatment  of,  55, 
225;  written,  55-57 

Exhibition,  or  "red  letter  day"  lesson, 
method  of  conducting,  136;  ob- 
ject of,  135 ;  place  for,  135 ;  prepar- 
ation for,  135-136;  program  of, 

136-137 
Existence,  struggle  for,  169-170,  171, 

1 80 

Experiences,  34,  58 
Experiments,   how   to   conduct,    174- 

!75,  !  75~i  78;  home,  178,  186 
Explanations,  40 


236 


Index 


Fabre,  Henry,  158,  182 

Factoring,   8,    20,    22;    exercises   in, 

81-82;     game    of,    86-87;     lesson 

on,  80-82 
Factors,    highest    common,    20,    22; 

modifying,  25;    technical,  45 
Failures,  causes  of,  in  mathematics, 

3-4 

Federal  Bureau  of  Entomology,  159 

Figures,  rectilinear,  105,  109,  110-135 

Fiori,  29 

Fishes,  156,  160 

Flowers,  155,  177 

Fly,  house,  193 ;  tachina,  192 ;  ichneu- 
mon, 183,  191 

Flytrap,  196,  197 

Foods,  156 

Forests,  155 

Formulas,  algebraic,  31,  108;  arith- 
metic, 36,  38,  53 

Fractions,  20,  22-23,  3^-37,  63,  75; 
complex,  83-84 ;  definition  of  com- 
plex, 84;  lesson  on,  82-84;  multi- 
plication of,  82-83 

Frog,  156,  195 

Functions,  158 

Games,  ball,  33,  79,  86-87 

Geology,  31 

Geometricians,  lives  of,  136 

Geometry,  plane,  213;  applications 
of,  132;  bird's-eye  view  of  course 
in,  109;  deduction  of  a  proof  in, 
112-114;  discipline  of,  108;  divi- 
sions of,  105-106;  history  of,  107; 
meaning  of  the  word,  107,  109; 
originals  in,  127-128;  practical 
value  of,  107-109;  review  questions 
in,  109-110;  steps  taken  in  prov- 
ing a  proposition  in,  114;  study  of 
originals  in,  124-128;  suggestions 
for  studying,  114-116,  118 

Geometry,  solid,  141,  145-146 

Germination,  174;  experiments  in 
seed,  176-177 


Goethals,  George  W.,  159 

Grade,  testing  for  a  final,  98 

Grades,  arithmetic  in,  37 ;   supervised 

study  in,  146 
Graphite,  163 
Graphs,  20,  23,  38,  75 
Grasshopper,  157-158,  182;  dissection 

of  the,  187,  195 
Gravity,  specific,  217-218,  219 
Guidebooks,  textbooks  as,  161,  183 

Habitat,  158 

Hall-Quest,  Alfred  L.,  6,  14,  63 

Hamilton,  Sir  William,  28 

Handwriting,  65,  135 

Health,  157-158,  203;   restoration  of, 

73 

Herbarium,  177 
Heron  of  Alexandria,  28 
Home   work,    n,    57-58;    value   of, 

58,  186-187 
Hotz'  scales,  75 
House  fly,  193 

Ice  cream,  87-88 

Ichneumon  fly,  183,  191 

Index,  card,  143;  units  of,  7,  20,  47, 
51,  105,  144,  ISS-JSS,  213 

Insects,  156,  160,  182-183;  a  game 
about,  193-194;  beneficial,  183, 
189;  biting,  190;  characteristics 
of,  182,  189 ;  classification  of,  183, 
189;  economic  loss  from,  192; 
harmful,  183,  189,  193-194;  in- 
teresting incidents  concerning,  159, 
182;  list  of  supplementary  topics 
on,  192;  pictures  of,  194;  sucking, 
190 

Insurance,  casualty,  30 

Interest,  arousing  the  pupils',  160; 
problems  in,  38 ;  theory  of,  30  , 

Inventions,  29-30  .- 

Involution,  20,  23^ 

Iodine,  171 


Index 


237 


Iron, 163 
Italics,  use  of,  51 

Koch,  Dr.,  159 

Laboratory,  supervision  of  work  in, 
150,  210-223 

Lantern,  stereopticon,  197,  226 

Leaves,  155,  167 

Legibility,  importance  of,  65,  135 

Lesson,  aim  of  the,  71;  definitions  of 
various  types  of,  8-9;  private,  72; 
purpose  of  a  socialized,  88 

Lesson  types:  correlation,  34-43, 
189-192,  204-206;  deductive,  117- 
122,  129-133,  202-204;  deductive 
and  how  to  study,  82-84,  110-116; 
examination,  95-101,  206-209;  ex- 
pository and  how  to  study,  73-80, 
92-95,  214-218;  how  to  study, 
43-50,  123-129,  161-165,  171-175. 
181-187,  197-199,  223-225;  in- 
ductive, 59-67,  67-73,  165-168, 
169-171,  175-178,  225-227;  induc- 
tive and  how  to  study,  50-59 ; 
laboratory,  187-189,  200-202,  219- 
223;  preview,  inspirational,  26- 
34,  106-110,  156-160;  red  letter 
day  (see  Program),  85-88,  135- 
J37>  I9S~I97»  socialized,  80-82, 
179-181,  192-194;  socialized  re- 
view, 88-91,  134-135 

Librarians,  144 

Library,  contents  of  biologic,  186 

Life,  biology,  the  study  of,  157 

Lincoln,  Abraham,  108 

Lines,  parallel,  105 

Loci,  105 

Logic,  108 

McMurry,  Frank  M.,  45 
Magazines,  34,  160,  186,  226,  227 
Mammals,  156,  160 
Man,   existence  of,    157;    study  of, 
160, 167 


Manuals,  laboratory,  151-152 

Material,  accumulation  of,  55,  179- 
181,  186 ;  available,  155 ;  exami- 
nation of  biologic,  161 ;  importance 
of  a  variety  of,  167;  source  of,  45, 
50,  79,  149,  160,  178-179;  supple- 
mentary, 14,  45,  80,  132,  151,  165; 
treatment  of,  53-55,  150,  167 

Mathematicians,  pictures  of,  27 

Mathematics,  characteristics  of,  3; 
contributors  to,  28;  English  of, 
27;  failures  in,  3;  language  of,  52; 
mastery  of,  71 ;  practical  value  of, 
29-32 ;  severity  of,  4,  224 

Matter,  functions  of  living,  158 

Memoranda,  14,  62,  130,  144 

Memorization,  kinds  of,  46;  power 
of,  5;  reliance  on,  116 

Memory,  employment  of,  121,  224; 
function  of,  46 ;  overdeveloped,  4 

Mensuration,  108 

Meteorology,  31 

Method,  adaptation  of,  59 ;  Austrian, 

37 

Microscope,  use  of,  188-189,  200-201 

Milkweed,  181 

Mimeograph,  use  of,  48,  73,  77,  127, 
151-152 

Mistakes,  common,  65;  correction 
of,  60,  65,  128-129,  132,  169;  ex- 
amples of  common,  38 ;  glaring,  65 ; 
how  to  avoid,  65-66;  how  to  find, 
84,  94,  214 

Monitors,  pupils  as,  92 

Monocotyledon,  derivation  of  word, 
172 

Monomials,  addition  of,  21,  61,  68-70; 
division  of,  22 ;  factoring,  22, 
multiplication  of,  21 ;  subtraction 
of,  21 

Morale,  41,  64 

Morey,  158 

Moritz,  R.  E.,  30-31 

Mosaics,  108 

Mosquito,  183,  186,  193 


238 


Index 


Mountains,  169 
Moving-picture  machine,  226 
Multiples,  common,  20,  22 
Muscles,  156,  198;    biologic  effect  of 

exercise  on,  204-206 ;   questions  on, 

199 

Museum,  making  a,  178 
Myers,  G.  W.,  9 

Narcotics,  156 

Nature,  dissimilarities  in,  167;  tran- 
quillity of,  169 
Naval   Observatory,   at   Washington, 

3i 

Navigation,  31 

Neatness,  importance  of,  135 

Newspapers,  34,  160,  186 

Newton,  Sir  Isaac,  28,  35 

New  York  State  Education  Depart- 
ment, lantern  slides  from,  197 ; 
statistics  of,  3,  20;  syllabus  of,  20, 
iSS-iS^ 

Notebooks,  criticism  of  the  work  in, 
133;  display  of  the  best,  133,  195- 
196;  how  to  study  the  returned, 
132;  loose-leaf,  15,  162;  manip- 
ulation of,  128-129,  130-132,  162, 
184;  pupils',  125-126;  record- 
ing experiments  in,  164-165,  168, 
174-177,  179,  219-222;  the  teach- 
er's, 34;  value  of,  129 

Notes,  historical,  34,  105 

Numbers,  literal,  21 ;  positive  and 
negative,  20,  21,  60,  61 ;  signed,  21 

Operations,  fundamental,  34,  36 
Originals,    rules    for    studying,    128; 

study  of,    125-128 
Outline,  for  the  study  of  insects,  194 

Panama  Canal,  159 

Paraffin,  218 

Paragraph,  study  of  the,  51-52,  54, 

1 88,  198;   writing  a,  166 
Paramecium,  195 


Parentheses,  21 ;   removing  of,  38,  86 

Parthenogenesis,  192 

Patterns,  tile,  108 

Payments,  equation  of,  30 

Percentage,  8,  36 

Period,  length  of,  10-11,  68,  150; 
management  of,  17-18 

Philosophy,  school  of,  108 

Phosphorus,  163,  165 

Photography,  222 

Photos,  226 

Physician,  31,  73,  186,  226 

Physics,  31,  213,  222,  223;  lantern 
slides  on,  226;  teacher  of,  226 

Physiography,  213 

Physiology,  31,  155-156,  195 

Picnic,  an  educational,  87-88 

Plant,  electric  power,  226 

Plants,  cellular  structure  of,  155 ; 
classification  of,  171;  problems  of, 
170;  study  of,  157,  i?7 

Plumule,  172 

Pointer,  use  of  the,  124 

Polygons,  areas  of,  105-106;  regular, 
105-106 ;  similar,  105-106 

Polynomials,  68;  addition  of,  21,  70- 
71;  division  of,  22;  factoring,  22; 
multiplication  of,  21 ;  rule  for 
adding,  71 ;  subtraction  of,  21 

Postulates,  105 

Potato  bug,  193 

Press,  hydrostatic,  226 

Preview,  inspirational,  26,  34,  106- 
no,  141,  156-160,  182;  conditions 
for  a  successful,  33-34;  in  arith- 
metic, 60-61 ;  method  of,  26,  157- 
160;  need  of,  26;  purpose  of,  26, 
106-107,  156-157;  questions  on, 

35 

Principal,  136 

Prizes,  85 

Problems,  analysis  of  several,  78 ;  ap- 
plying the  equation  to,  77 ;  business, 
30-31 ;  conception  of,  72 ;  daily 
study  of,  79,  166-167;  different 


Index 


239 


kinds  of,  96-97,  223;  directions 
for  studying,  77-78,  127,  223-224; 
questions  leading  up  to  the  study 
of,  75 ;  recognition  of,  61-62,  70- 
71;  sensing,  45,  168;  solving,  36; 
standardized  tests  on  problems, 
75;  statement  of,  112 

Program  (see  Lessons,  types  of) :  pur- 
pose of,  195;  "red  letter  day," 
85-88,  136-137,  I9S-I97 

Program  of  studies,  2 ;  place  of  ad- 
vanced mathematics  in,  141 

Proportion,  105-106 

Protractor,  use  of,  137 

Psychology,  31 

Pupils,  characteristics  of,  13,  72-73 ; 
classification  of,  16,  18,  98;  collec- 
tion of  specimens  by,  160;  criti- 
cism of  work  by,  203 ;  elimination 
of,  13,  69,  70;  embarrassment  of, 
72;  free  expression  of,  34;  grading 
of,  1 8;  grouping,  72;  guidance  of, 
13 ;  ingenuity  of,  152 ;  judgment  of, 
82 ;  judgment  on  part  of,  135,  191 ; 
maximum,  137,  189;  name  of,  14; 
responsibility  of,  177,  184;  seating 
of,  17,  42,  72;  self-reliance  of,  123, 
185  / 

Pyramids,  107,  109 

Pythagoras,  107,  no 

Quadrilaterals,  105 

Quartz,  218 

Quaternion  Bridge,  29 

Questions,  how  to  ask,  187,  191 ;  im- 
portance of  asking  skillful,  131 

Quiz,  importance  of  oral,  6,  166; 
method  of  the,  200,  202-203 

Race,  championship,  85 ;  division,  85 ; 

multiplication,  85 ;    relay,  86 
Radicals,  20,  24,  75,  88 
Reading,  necessity  of  intelligent,  78; 

supplementary,  151,  186 
Reasoning,  faculty  of,  5 ;  undeveloped 

powers  of,  4 


Recitation,  the  complete,  205;  the 
unsupervised,  5 ;  types  of,  8 ;  units 
of,  8,  21-24,  5i,  105-106,  165,  213 

Record  of  work,  14,  16 

Recreation,  15 

Regents  academic  examinations,  3 

Resourcefulness,  of  pupils,  69,  152,  208 

Results,  checking,  18,  66-67 

Review,  methods  of,  11-12,  35,  44, 
50-51,  59-60,  68-70,  73-75,  80-81, 
82,  83,  88-91,  92,  uo-iii,  117-118, 
123-125,  129-132,  134-135,  165- 
166,  169,  171,  175,  181-182,  187, 
189-191,  192,  193-194,  197-198, 

200,       202-203,      204-205,      214-215, 

219,  223;  nature  of  the,  41,  46,  161, 
175;  purpose  of,  n;  socialized, 
60,  88-91,  134-135 ;  summary  of 
the,  40-41 

Roll  call,  17,  67 

Romanes,  George,  159,  182 

Romans,  29,  35 

Roosevelt  Dam,  108 

Roots,  155 

Rugg  and  Clark's  tests,  74-75 

Ruler,  44 

Sanitation,  156 

Schedule,  daily  lesson,  amplification 
of,  68 ;  divisions  of,  10-1 1 ;  impor- 
tance of,  17;  sample  sheets,  19,  62; 
time  (see  Lessons,  types  of) 

School,  the  average,  122;  the  cor- 
respondence, 144 

Schoolroom,  9,  144 

Schultze,  Arthur,  4 

Science,  algebra,  a  general,  36;  fas- 
cination of,  214;  function  of  the 
study  of,  205 ;  importance  of  super- 
vised study  in,  150,  173;  popularity 
of,  149;  practical  aspect  of,  149; 
the  study  of,  149 

Scorekeeper,  87 

Seeds,  155,  171-175,  176-178;  dis- 
persal ofj  181 


240 


Index 


Semester,  141 

Sheets,  mimeograph,  48,  73,  77,  127, 
151-152 

Shipbuilding,  30 

Signs,  72 

Simpson,  Mabel  E.,  13,  59 

Smith,  Dr.  Eugene,  29 

Snapdragon,  181 

Solar  system,  108 

Spheres,  109 

Square,  method  of  completing  the,  93 ; 
rule  for  completing  the,  93;  the 
perfect,  86-87 

Squash  bug,  193 

Standardized  tests,  6,  97-98;  Hotz', 
75 ;  the  Rugg  and  Clark's,  74,  97 ; 
value  of,  74 

Statements,  loose,  199 

Station,  pumping,  226 

Statistics,  39 

Stems,  155 

Stenographer,  58 

Stereopticon,  197,  226 

Stereoscopic  views,  145 

Stimulus,  supplying  the  proper,  70 

Strayer,  George  D.,  8 

Study  period,  function  of,  12-13; 
management  of,  17,  39,  43,  63-67; 
organization  of,  46-47 ;  the  logical 
culmination  of,  146;  the  teacher's 
duty  during  the,  38,  65-66,  118; 
work  done  outside  of  the,  57-58 

Study,  how  to,  instruction  in  (see  also 
how  to  study  lessons),  43-49,  5I~55> 
78,  114-116,  127,  128,  131,  161-165, 
172-173,  184-198,  214-215;  pur- 
pose of  lessons  on,  43-44,  161 

Study,  cooperative,  57;  correct  habits 
of,  54;  methods  of,  44-46,  114-116, 
110-122,  163-165,  188-189;  sum- 
mary on  the  silent,  43;  the  period 
of  silent,  39-40.  63-66,  84,  94~95> 
116,  110-122,  133,  184-186,  1 88- 
189 ;  units  of,  8 ;  value  of  outside, 
57-58, 186-187 


Studying,  cooperative,  57 ;  rules  for,  49 

Submarine,  218 

Sulphur,  163-165 

Superintendent,  136 

Supervised   study,   function  of,   5-6, 

43-44,  69,  119,  151;  Hall-Quest  on, 

xiii-xvi;     installation     of,     10-11; 

management   of,  17;     meaning   of, 

149;     organization   of,   46-47,   68; 

relationship  of,  6;    technic  of,  6; 

value  of,  5,  42,  150,  173 
Surveying,  31,  107,  no,  137 
Symbols,  21,  35,  54;  meaning  of,  53; 

origin  of,  39 
Symmetry,  106 
Sympathy,  6,  12,  13,  43,  96,  146 

Table,  time,  25,  156 

Tachina  fly,  192 

Tartaglia,  29 

Teacher,  205 ;  activity  of,  80 ;  check- 
ing up  work  by,  66-67;  duty  of, 
40,  65-66,  121,  124,  131,  144,  184, 
189,  245;  helps  for,  63-64;  judg- 
ment of,  33,  134;  leadership  of, 
152;  opportunity  of,  168;  origi- 
nality and  individuality  of,  59; 
preparatory  work  of,  171-173,  179- 
180;  tactfulness  of,  72;  the  pupil 
as  teacher,  198 ;  use  of  standardized 
tests  by,  74-75 

Technic,  mastery  of,  95,  146 

Terms,  significance  of,  72 ;  transposi- 
tion of,  28,  76-77,  93,  94;  use  of,  6 

Tests,  6,  142,  166;  Hotz',  75;  prob- 
lems as  real,  96-97 ;  real,  95-97 ; 
Rugg  and  Clark's,  74,  97;  speed, 
73-74;  standardized,  6,  97-98; 
time,  45 ;  written,  97-98 

Textbooks,  109;  arithmetic,  38; 
study  of,  47,  149,  183-184;  sup- 
plementary, 51,  57-59,  91,  I22, 
186-199;  use  of,  44,  161,  203; 
varying  characteristics  of,  167 ; 
verification  of,  177,  223 


Index 


241 


Thales,  30,  137 

Theorem,  112,  117;  explanation  of 
the  new,  118;  review  of  the,  134- 

i3S 

Thoroughness,  importance  of,  135, 
144,  165,  174 

Time  (see  also  each  lesson  outlined), 
allotment  of,  17,  150;  amount  of, 
59,  150;  efficient  use  of,  6,  174-175 

Toad,  value  of,  186,  192 

Tools,  44,  152 

Triangles,  105,  109,  123;  definition 
of,  118 

Trigonometry,  plane,  141,  145-146 

Type  forms,  explanations  of,  61 ;  dif- 
ferent, 82 

Types,  classification  of  exercise,  9 

Unknown,  use  of,  36,  76-78 


Variation,  meaning  of,  168 
Verification,     importance     of,     149; 

methods   of,    66-67,   72-73,  93~94, 

1 20,  176,  202 
Vieta,  28,  35 
Volume,  217-218 

Weber-Fechner  law,  31 
Weeds,  170,  186 
White,  C.  E.,  30 
Wiley,  Dr.  Harvey,  159 
Wings,    classification    of    insects    ac- 
cording to,  100 
Work  bench,  152 

X-ray,  226 

Zoology,  155,  183,  195 


SOUTHERN  BRANCH, 

UNIVERSITY  OF  CALIFORNIA, 

LLBKARY, 
IU>S  ANGCLES,  CALIF. 


